BY ABOLISHING IN HIS FLESH THE ENMITY: katargesas, (AAPMSN): (note that the phrase "in His flesh" "en sarki auto" is found in the Nestle-Aland Greek text at the end of verse 14) (Colossians 1:22; Hebrews 10:19-22)

By abolishing in His [own crucified] flesh the enmity [caused by] the Law with its decrees and ordinances [which He annulled]; that He from the two might create in Himself one new man [one new quality of humanity out of the two], so making peace. (Amplified)

Abolishing modifies "broke down" in Ephesians 2:14 (note).

Abolishing (2673) (katargeo from kata = intensifies meaning + argeo = be idle from argos = ineffective, idle, inactive from  a = without + érgon = work) (Click word study on katargeo) literally means to reduce to inactivity. The idea is to make the power or force of something ineffective and so to render powerless, null and void. To cause something to come to an end or to cease to happen. The aorist tense in this context depicts a once for all completed action in the past.

The word abolish simply means “to nullify.” The Law no longer holds sway over either Jew or Gentile, since in Christ believers are not under Law but under grace.

Hughes writes asks how did Christ abolish the law

How did he do this, especially since he said in his Sermon on the Mount, “Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them” (see note Matthew 5:17)? Christ fulfilled the moral law, keeping all its requirements, but he abolished the Jewish ceremonial law. Thus, the requirements of the ceremonial law (the washings, the Sabbath restrictions, etc.) which had been such a barrier were gone. And since he fulfilled the moral law, taking away its condemnation, all have free access through grace (cf. 2 Corinthians 3:6-15). The gospel is now, “For it is by grace you have been saved, through faith — and this not from yourselves, it is the gift of God — not by works, so that no one can boast” (Eph 2:8, 9), and because of this we fly across the barrier to God! (Hughes, R. K.: Ephesians: The Mystery of the Body of Christ. Crossway Books)

The Net Bible note adds that "abolishing" can be translated...

“rendered inoperative.” This is a difficult text to translate because it is not easy to find an English term which communicates well the essence of the author’s meaning, especially since legal terminology is involved. Many other translations use the term “abolish” (so NRSV, NASB, NIV), but this term implies complete destruction which is not the author’s meaning here. The verb katargeo can readily have the meaning “to cause something to lose its power or effectiveness”, and this meaning fits quite naturally here within the author’s legal mindset. A proper English term which communicates this well is “nullify” since this word carries the denotation of “making something legally null and void.” This is not, however, a common English word. An alternate term like “rendered inoperative [or ineffective]” is also accurate but fairly inelegant. For this reason, the translation retains the term “nullify”; it is the best choice of the available options, despite its problems. (The NET Bible Notes. Biblical Studies Press)

Flesh (4561) (sarx) (Click word study on sarx)

In His flesh - the physical death of Christ, which the Father made possible because...

when the fulness of the time came, God sent forth His Son, born of a woman, born under the Law. (Galatians 4:4) (Comment: The Divine Son of God was also the Son of Man, born by the Spirit of a woman and so fully God and fully Man, His perfect Humanity being necessary so that He could abolish the enmity in His physical flesh and free us from our sins by the New Covenant in His blood.)

Barnes writes...

By the sacrifice of his body on the cross. It was not by instruction merely; it was not by communicating the knowledge of God; it was not as a teacher; it was not by the mere exertion of power; it was by his flesh--his human nature--and this can mean only that he did it by his sacrifice of himself. It is such language as is appropriate to the doctrine of the atonement--not indeed teaching it directly--but still such as one would use who believed that doctrine, and such as no other one would employ. Who would now say of a moral teacher that he accomplished an important result by his flesh? Who would say of a man that was instrumental in reconciling his contending neighbors, that he did it by his flesh?... No man would have ever used this language who did not believe that Jesus died as a sacrifice for sin. (Albert Barnes. Barnes NT Commentary).

Enmity (2189) (echthra from echthros = enemy, hostile) (Click word study on echthra) means hostility, a reason for opposition, enmity, hatred. Enmity suggests positive hatred which may be open or concealed whereas hostility suggests an enmity showing itself in attacks or aggression.

Echthra is that spirit that looks with evil suspicion on anyone of a different race, tongue, nation, or creed. It is the “attitude of heart and mind that puts up barriers and draws the sword,” but Christ has broken down the barrier and has taught us to love those who are “hostile”.

Here enmity refers to the personal and national prejudice and exclusiveness between Jews and Gentiles a result primarily of the separating influence of the Mosaic legal system. Christ abolished this at Calvary effecting a great reconciliation and uniting hostile members of the human family. Christ, the prophesied Prince of peace, is the world’s only hope of lasting peace! Let us therefore "pray for the peace of Jerusalem", realizing this is a prayer in essence for our Lord to return. Maranatha!

Regarding enmity Barnes explains that...

The idea is, that the ceremonial law of the Jews, on which they so much prided themselves, was the cause of the hostility existing between them. That made them different people, and laid the foundation for the alienation which existed between them. They had different laws; different institutions; a different, religion. The Jews looked upon themselves as the favorites of Heaven, and as in possession of the knowledge of the only way of salvation; the Gentiles regarded their laws with contempt, and looked upon the peculiar institutions with scorn. When Christ came, and abolished by his death their peculiar ceremonial laws, of course the cause of this alienation ceased.  (Albert Barnes. Barnes NT Commentary)

John Eadie writes that the enmity between Jew and Gentile was...

"hatred which rose like a party wall, and kept both races at a distance. Deep hostility lay in their bosoms; the Jew looked down with supercilious contempt upon the Gentile, and the Gentile reciprocated and scowled upon the Jew as a haughty and heartless bigot. Ample evidence is afforded of this mutual alienation. Insolent scorn of the Gentiles breaks out in many parts of the New Testament (Acts 11:3, 22:22; 1Thess. 2:15), while the pages of classic literature show how fully the feeling was repaid. This rancor formed of necessity a middle wall of partition, but Jesus, Who is our Peace, hath broken it down. (Eadie, J: Ephesians)

WHICH IS THE LAW OF COMMANDMENTS CONTAINED IN ORDINANCES: ton nomon ton entolon en dogmasin katargesas, (AAPMSN):  (Galatians 3:10; Colossians 2:14,20; Hebrews 7:16; 8:13; 9:9,10,23; 10:1-10)

Note that  order of words in the Greek is "the enmity in his flesh, the law of the commands in ordinances having done away" which signifies that our Lord abolished the Law by His death on the Cross.

Law (3551) (nomos) in its primary meaning relates to that which is conceived as standard or generally recognized rules of civilized conduct.

Vincent has this note...

The law, etc., depends in construction on having abolished, and is not in apposition with the enmity, as A. V. The middle wall of partition, the enmity, was dissolved by the abolition of the law of commandments. Construe in His flesh with having abolished. Law is general, and its contents are defined by commandments, special injunctions, which injunctions in turn were formulated in definite decrees. Render the entire passage: brake down the middle-wall of partition, even the enmity, by abolishing in His flesh the law of commandments contained in ordinances (Ephesians 2)

Commandments (1785) (entole from entellomai = order, give commandments) is most common of the words meaning commandment, stressing the authority of the one commanding, while éntalma (G1778), a religious commandment, stresses the thing commanded.  It refers to law in general.

Contained is added by the translators for continuity.

Ordinances (1378) (dogma from dokéo = to think) refers to a fixed and authoritative decision or requirement (see the "decree" [dogma] of the emperors in Lu 2:1, Acts 17:7).

NIDNTT writes that in classic Greek dogma...

stems from the verb dokeo (think, suppose, imagine, conclude), and means opinion, conclusion, belief. It occurs only 3 times in the pre-Socratic writers and always in connection with Pythagoras. From Xenophon onwards in the fourth cent. B.C. it has the following meanings: (1) opinion (in ordinary speech); (2) a doctrine (in philosophy, e.g. Epicurus, De rerum natura, 14, 1, 15 and 28); (3) a decree of God (in religious writers); (4) a decree, ordinance, edict (in official language, with the emphasis on public promulgation). (See Arndt, 200.)...Where dogma is used in a general, secular sense as an official, public decree, it has only indirect theological significance...With the death of Christ, the law with all its commandments and ordinances “is removed from the world as a factor in salvation”...The use of dogma is, therefore, all the more surprising when used in a positive sense (cf. above OT) for teaching that is binding on the whole church. Acts 16:4 lays the foundation for the idea of dogma as an ecclesiastical decree, requiring intellectual assent. It runs the risk of turning the gospel of Christ into legalism. On the other hand, the pressure from the Judaizers forced the Jerusalem council to take a stance in defining their attitude. The dogmata of the council were in fact decrees proclaiming liberty within a defined area rather than a series of tight restrictions. (Brown, Colin, Editor. New International Dictionary of NT Theology. 1986. Zondervan)

TDNT sums up dogma...

The basic meaning is “what seems to be right”: a. “opinion,” b. “principle,” c. “resolution,” d. “decree,” and e. “the law.” The verb means “to affirm an opinion,” “to establish a decree,” “to publish an edict.”
1. In the NT sense d. occurs in Lk. 2:1; Acts 17:7; Heb. 11:23.
2. In Colossians 2:14 (note) the reference might be to the new edict of God but in 2:20 we definitely have legal ordinances (sense e.), so that the real point in 2:14 is that Christ has canceled these. Eph. 2:15 carries a similar reference to the ordinances of the law.
3. In Acts 16:4 the term is used for the resolutions of the apostolic council. The apostolic fathers then adopt the term for the teachings of Jesus. (Kittel, G., Friedrich, G., & Bromiley, G. W.  Theological Dictionary of the New Testament. Eerdmans)

Dogma refers to a formal statement concerning rules or regulations that are to be observed -- the idea is a formalized sets of rules which might refer to an ordinance, a decision or a command. This is the primary meaning of dogma in Ephesians 2:15. Dogma thus refers to the rules and requirements of the law of Moses, in this verse specifically referring to the "ceremonial laws" or ordinances covering the various aspects of the Jewish feasts, sacrifices, offerings, laws of cleanliness and purification, and all other such distinctive outward commandments for the unique separation of Israel from the nations. Paul is saying that "in His flesh" on the Cross, Jesus abolished or made to no effect these dogma

Dogma is used  of certain decrees of the apostles relative to right living (see Acts 16:4)

Dogma can refer to something that is taught as an established tenet or statement of belief (dogma).

Dogma was especially used in ancient Rome to describe the public decrees  of the Roman Senate.

Dogma is used 5 times in the NT (Luke; Acts 2x; Ephesians; Colossians) and is translated: decree, 1; decrees, 3; ordinances, 1. Dogma is found in the Septuagint (LXX) only in Daniel (Da 2:13; 3:10, 12, 29; 4:6; 6:8ff, 12f, 15, 26)

The law consisted of decrees or commands. Dogma is used for God’s laws and the external precepts of the Mosaic Law. It referred to a legal obligation which was a binding law or edict which was placed on a public place for all to see.

In English, dogma means something held as an established opinion; a definite authoritative tenet; code of authoritative tenets; doctrine or body of doctrines concerning faith or morals formally stated and authoritatively proclaimed by a church -- if one is "dogmatic" he is unduly and offensively positive in laying down principles and expressing opinions

In the late Judaism of the first century AD, Philo and Josephus understood the Mosaic law as a system of holy tenets, referred to as the dogmata of a divine philosophy. As the most exalted of all systems, it was superior to the doctrines of the rest of ancient philosophy.

The Law of Moses was a single legislative code which was in turn composed of separate, formal commandments, which in turn consisted of dogmas or decrees covering many, if not most, areas of life.

The Law did set up Israel as God’s chosen earthly people, but unfortunately many Jews became arrogant and treated Gentiles with contempt. The Gentiles responded with deep hostility, known all too well as anti-Semitism. And yes, it still exists even in the Church of Jesus Christ!.

Wayne Barber explains what was abolished writing that...

The Law was divided into the moral law and the ceremonial law. He didn’t abolish the moral law. That has always been here and is fulfilled when we obey the Lord Jesus Christ (eg, see notes Matthew 5:18; 5:19). The moral law says that we love God with all of our heart and our mind and our strength, and we are to love one another (see notes Romans 13:9; 13:10). That is always there and is morally built in. He did not make that obsolete. He did not make that ineffective. What He did do was to render ineffective the ceremonial law. It says here, the "commandments contained in ordinances". In other words, what He did was put religion to death. No longer could the Jew say,

"Oh, I sacrifice. I go to the Temple. I worship on the Sabbath. I do this. I do that. God loves me more than He loves you."

Oh, no. He took all the external stuff and threw it out. He says, "Now there is only one way to God, and that’s through Me. You can’t work your way up the ladder." (Ephesians 2:15-18 Christ the Author of Our Peace - 2)

John MacArthur has some instructive comments on this section writing that...

The greatest barrier between Jew and Gentile was the ceremonial law, the Law of commandments contained in ordinances. The feasts, sacrifices, offerings, laws of cleanliness and purification, and all other such distinctive outward commandments for the unique separation of Israel from the nations were abolished. That God’s moral law was not abolished is clear from the phrase contained in ceremonies. His moral law reflects His own holy nature and therefore can never change (cf. Matt. 5:17-19)...All the ceremonial laws which distinguished and separated Jews from Gentiles were obliterated. Before Christ those groups could not eat together because of restricted foods, required washings, and ceremonial contamination. Now they could eat anything with anyone. Before Christ they could not worship together. A Gentile could not fully worship in the Jewish Temple, and a Jew would not worship in a pagan temple. In Christ they now worshiped together and needed no temple or other sacred place to sanctify it. All ceremonial distinctions and requirements were removed (cf. Acts 10:9-16; 11:17-18; Col. 2:16-17), (MacArthur, J: Ephesians. Chicago: Moody Press)

SO THAT IN HIMSELF HE MIGHT MAKE THE TWO INTO ONE NEW MAN: hina tous duo ktise (3SAAS) en auto eis ena kainon anthropon: (Colossians 1:22; Hebrews 10:19-22)  (Galatians 3:10; Colossians 2:14, 20; Hebrews 7:16; 8:13; 9:9,10,23; 10:1-10)  (4:16; 2 Corinthians 5:17; Galatians 6:15; Colossians 3:10)

So that (2443) (hina) introduces a purpose clause.

In Himself - He is the medium or means of reconciliation.

Make - more literally "create". This work was a new creation on a new foundation with the cornerstone being Christ Himself.

He might make (2936) (ktizo) was a word meaning to create something out of nothing such as God in the act of creation of the universe, but in the present context referring to a spiritual creation, the church, an entity that had never existed prior to this time. In fact, there is no mention of the church in the Old Testament, although many commentators who espouse a non-literal interpretation of Scripture, have wrongly interpreted many of the promises God specifically gave to Israel as being applicable to the church.

Paul had just used the same verb, ktizo, pointing to the saints, each one representing God's new creation in Christ.

For we are His workmanship, created (ktizo) in Christ Jesus for good works, which God prepared beforehand, that we should walk in them. (see note Ephesians 2:10)

Paul uses ktizo two more times in Ephesians, the first referring to God's creation of the universe...

To me, the very least of all saints, this grace was given, to preach to the Gentiles the unfathomable riches of Christ, and to bring to light what is the administration of the mystery which for ages has been hidden in God, who created (ktizo) all things (see notes Ephesians 3:8, 3:9)

(Paul instructs the Ephesian saints to) put on the new self, which in the likeness of God has been created (ktizo) in righteousness and holiness of the truth. (see note Ephesians 4:24)

Two (1417) (duo) is the cardinal number 2 here referring to the "duo" of Jew and Gentile now joined together.

One (1520) (heis) is the cardinal numeral one and in this verse defines that which is united as one in contrast to being divided or consisting of separate parts. Heis speaks of oneness, unity and identity, Jew and Gentile united in position and privilege. Now race and national distinctions disappear as Paul explained to the Galatians writing that...

There is neither Jew nor Greek (Gentile), there is neither slave nor free man, there is neither male nor female; for you are all one (heis) in Christ Jesus. (Galatians 3:28)

New (2537) (kainos) means new in kind or quality, unprecedented, unheard of, new in sense that it brings into the world a new quality of thing which did not exist before.

The Net Bible Notes explains that...

In this context the author is not referring to a new individual, but instead to a new corporate entity united in Christ... This is clear from the comparison made between the Gentiles and Israel in the immediately preceding verses and the assertion in Ephesians 2:14 that Christ “made both groups into one.” This is a different metaphor than the “new man” of Eph 4:24; in that passage the “new man” refers to the new life a believer has through a relationship to Christ. (The NET Bible Notes. Biblical Studies Press)

Kainos signifies qualitatively new in contrast to néos which indicates temporally new or new with respect to age. Neos is new simply in point of time; a thing which is neos has come into existence recently, but there may well have been thousands of the same thing in existence before. A pencil produced in the factory this week is neos, but there already exist millions exactly like it. Kainos on the other hand is new in point of quality, new in sense that it brings into the world a new quality of thing which did not exist before.

In the present context the thing which had never existed was believing Jew and believing Gentile together as one entity.

In rabbinic Judaism,  a pagan Gentile coming to know God is thought of as if he had been created by whoever helped him to attain knowledge of God

One new man - Corporately not individually. In other words, this truth in this context refers to Christ's body, the church, which is in turn composed of individual new creations in Christ (cf 2Cor 5:17). Note that God is not making a new world, but a new man. God makes no attempt to improve world conditions by repairing the old systems, but He replaces the old, earthly nationalisms by a new order whose citizenship is of heaven.

 Paul summed up this new entity (one new man)  when he said,

“For there is no distinction between Jew and Greek; for the same Lord is Lord of all, abounding in riches for all who call upon Him; for ‘Whoever will call upon the name of the Lord will be saved’ ” (Ro 10:12-13).

MacDonald explains that...

 The church is new in the sense that it is a kind of organism that never existed before. It is important to see this. The NT church is not a continuation of the Israel of the OT. It is something entirely distinct from anything that has preceded it or that will ever follow it. This should be apparent from the following:

1. It is new that a Gentile should have equal rights and privileges with a Jew.
2. It is new that both Jews and Gentiles should lose their national identities by becoming Christians.
3. It is new that Jews and Gentiles should be fellow members of the Body of Christ.
4. It is new that a Jew should have the hope of reigning with Christ instead of being a subject in His kingdom.
5. It is new that a Jew should no longer be under the law.

The church is clearly a new creation, with a distinct calling and a distinct destiny, occupying a unique place in the purposes of God. (MacDonald, W & Farstad, A. Believer's Bible Commentary: Thomas Nelson)

Clement of Alexandria wrote

“We who worship God in a new way, as the third race, are Christians.”

The Epistle of Diogenes calls believers “this new race.”

In Christ there is no East or West,
In Him no South or North,
But one great fellowship of love
Throughout the whole wide earth.
John Oxenham

Chrysostom gives a striking illustration...

"Let us imagine that there are two statues, one of silver and the other of lead, and then that both shall be melted down, and the two shall come out gold. So thus He has made the two one.”

Eadie explains one new man this way...

One new man—both races being now enabled to realize the true end of humanity; Gentile and Jew not so joined that old privilege is merely divided among them. The Gentile is not elevated to the position of the Jew—a position which he might have obtained by becoming a proselyte under the law; but Jew and Gentile together are both raised to a higher platform than the circumcision ever enjoyed. The Jew profits by the repeal of the law, as well as the Gentile. Now he needs to provide no sacrifice, for the One victim has bled; the fires of the altar may be smothered, for the Lamb of God has been offered; the priest, throwing off his sacred vestments, may retire to weep over a torn veil and shattered temple, for Jesus has passed through the heaven “into the presence of God for us;” the water of the “brazen sea” may be poured out, for believers enjoy the washing of regeneration; and the lamps of the golden candelabrum have flickered and died, for the church enjoys the enlightening influences of the Holy Spirit. Spiritual blessing in itself, and not merely pictured in type, is possessed by the Jew as well as the Gentile. (John Eadie, D., LL.D. The Epistle of St Paul to the Ephesians)

 THUS ESTABLISHING PEACE: poion (PAPMSN) eirenen: (4:16; 2 Corinthians 5:17; Galatians 6:15; Colossians 3:10)

Establishing (4160) (poieo) means to make or produce. The present tense speaks of the continual effect to make peace between Jew and Gentile at all times and seasons because the barriers that separated them have been torn down at the Cross. All are on equal footing at the foot of His Cross.

Peace (1515) (eirene from the verb eiro = to bind or join together what is broken or divided) (Click word study on eirene) means in essence to set at one again or join together that which is separated. In secular Greek eirene described the cessation or absence of war. Christ has established peace between Jew and Gentile by removing the cause of hostility, by imparting a new Spirit indwelt nature, and by creating a new union, the body of Christ. The Cross of Christ is God’s answer to racial discrimination, segregation, anti-Semitism, bigotry, and every form of strife between men. Paul relates this same truth in Colossians affirming that the saints have...

put on the new self who is being renewed to a true knowledge according to the image of the One who created him--a renewal in which there is no distinction between Greek and Jew, circumcised and uncircumcised, barbarian, Scythian, slave and freeman, but Christ is all, and in all. (see notes Colossians 3:10, 3:11) - see especially

"Christ, as the all-sufficient Lord and Savior, is all that matters. His Spirit-mediated indwelling in all believers, of whatever racial-religious, cultural, or social background they be, guarantees the creation and gradual perfection in each and in all of “the new man, who is being renewed for full knowledge according to the image of him who created him.” Thus, most appropriately, the very theme of the entire letter, namely, “Christ, the Pre-eminent One, the Only and All-Sufficient Savior,” climaxes this passage." (Hendriksen, W., & Kistemaker, S. J. New Testament Commentary Set, 12 Volumes. Grand Rapids: Baker Book House)

Wayne Barber has an interesting discussion of what Jesus did when He abolished "in His flesh the enmity"...

The word "abolished" is katargeo. That is the word that means to make useless, to render ineffective. He gave them a brand new way. Jesus abolished the Law.

He said it was an enmity. The word "enmity" here in this context means the cause of enmity. What was the cause of enmity between the Jew and the Gentile? It was their Laws and their observances, which they thought made them more spiritual than anybody else and had become their source of pride. Jesus put an end to the cause of the hatred that existed between the Jews and the Gentiles. How did He do it?

It says, "by abolishing in His flesh the enmity" There are two things that are brought into that.

First, by living a sinless life, Jesus fulfilled the Law, which no man could do. Once He fulfilled it, He was qualified to take it from there...

For what the Law could not do, weak as it was through the flesh, God did: sending His own Son in the likeness of sinful flesh and as an offering for sin, He condemned sin in the flesh, 4 in order that the requirement of the Law might be fulfilled in us, who do not walk according to the flesh, but according to the Spirit. (See notes Romans 8:3; 8:4)

Not only that, when He took sin upon Himself, He satisfied the curse of the Law. He became a curse for us.

"CURSED IS EVERYONE WHO HANGS ON A TREE " [Gal 3:13]"

The Scripture says. The curse of the Law was satisfied, therefore, rendering the Law ineffective when a person comes to Jesus Christ [Ro 7:1-6]. The person who rejects Jesus is guilty of all points of the Law. If a person comes to Christ, the Law has no effect whatsoever in his life to ever condemn him again. In Christ we find the fulfillment of that Law [Ro 8:3-4]. We find what we are looking for, that is the oneness that we need with God.