BBE:  And the women who were her neighbours gave it a name, saying, Naomi has a child; and they gave him the name of Obed: he is the father of Jesse, the father of David.
CEV:  The neighborhood women named him Obed, but they called him "Naomi's Boy." When Obed grew up he had a son named Jesse, who later became the father of King David.
GWT: The women in the neighborhood said, "Naomi has a son." So they gave him the name Obed. He became the father of Jesse, who was the father of David.
KJV: And the women her neighbours gave it a name, saying, There is a son born to Naomi; and they called his name Obed: he is the father of Jesse, the father of David.
NJB:  And the women of the neighbourhood gave him a name. 'A son', they said, 'has been born to Naomi,' and they called him Obed. This was the father of Jesse, the father of David. (NJB)
Young's Literal:  and the neighbouring women give to him a name, saying, 'There hath been a son born to Naomi,' and they call his name Obed; he is father of Jesse, father of David.

Septuagint (LXX):   kai ekalesan (3PAAI) autou ai geitones onoma legousai (PAPFPN) etechthe (3SAPI) huios te Noemin kai ekalesan (3PAAI) to onoma autou Obed houtos pater Iessai patros Dauid 

Here is the only place in the OT where a child was named by someone other than the immediate family. In a sense Ruth vicariously bore the son who would restore the family name of Naomi’s deceased son Mahlon (cf. Ruth 4:1).

SO THEY NAMED HIM OBED HE IS THE FATHER OF JESSE THE FATHER OF DAVID:

"Obed" (5744)  is transliterated from the Hebrew Oved which according to some sources means "serving" although other sources say Obed means "worshipper".   In either case, it is worth noting that names in the OT conveyed much more information about the individual than do most modern names.

This complete genealogy appears identically in four other biblical texts (Ru 4:21-22; 1 Chr 2:12-15; Mt 1:5, 6; Lu 3:31-32). Boaz and Ruth were the great grandparents of David. The story of Ruth has shown how a Moabite woman obtained an exalted place in Hebrew history. There is later evidence that David did not forget his Moabite roots. During the period of flight from Saul's wrath, David asked the king of Moab to let his parents stay there for refuge (1 Sam 22:3-4).

The following genealogical schematic is taken from The New Bible Dictionary (IVP, 1996)

G Campbell Morgan sums up the book writing that...

The story ends with poetic simplicity and beauty. "Boaz took Ruth, and she became his wife." Naomi at last was com­forted indeed. The women of her own people spoke words of cheer to her which unquestionably were full of comfort, as they set forth the praises of the one who had chosen to share her affliction, and had become the medium of her succour. There is a stately simplicity in this story of the issue. It constitutes a record of the Divine movement in the history of the chosen people, for thus the kingly line is ordained, m the midst of infidelity, through faithful souls. All the period of the Judges was characterized by the failure of the people to realize the great ideal of the Theocracy. They had no king because they were dis­obedient to the One King. Presently we shall hear them clamouring for a king "like the nations," and one will be ap­pointed by whose reign of forty years they will learn the difference between earthly rule and the direct government of God. Then the man after God's own heart will succeed him; and that man will be David, descended from these souls who, in dark and difficult days, realized in their own lives the Divine ideal, as they walked humbly with God. But this Book flings its light much further on. After centuries had run their course, there sprang from this union of Boaz and Ruth in faith and love the Man of Nazareth, Jesus, the One anti only King of men, because He was not only a Child born to Mary, descended from these, but also the Son of God in all the fulness of that title. God, in love and might, ever moves on through human failure, in co-operation with human faith. (Morgan, G. C. Life Applications from Every Chapter of the Bible).

BBE:  Now these are the generations of Perez: Perez became the father of Hezron;
CEV:  Here is a list of the ancestors of David: Jesse, Obed, Boaz, Salmon, Nahshon, Amminadab, Ram, Hezron, and Perez.  (CEV)
GWT: This is the account of Perez and his family. Perez was the father of Hezron. (GWT)
KJV: Now these are the generations of Pharez: Pharez begat Hezron,
NJB:  These are the descendants of Perez. Perez fathered Hezron, (NJB)
Young's Literal:  And these are genealogies of Pharez: Pharez begat Hezron,

Septuagint (LXX):   kai autai ai geneseis Phares Phares egennesen (3SAAI) ton Esron 

"Generation" (8435) (towledah; toledah) occurs only in the plural in the Hebrew (cf "generations") and describes one's descendants, generations or genealogies. This key Hebrew word carries with it the notion of everything entailed in a person’s life and that of his or her progeny. It is not surprising that Genesis records numerous uses of this word to introduce the idea of a "family tree".

For example, Moses records that

"This is the book of the generations of Adam. In the day when God created man, He made him in the likeness of God." (Genesis 5:1).

In the present case towledah/toledah introduces the genealogical list of the descendants of Ruth and Boaz. This recalls the prayer recorded in verse 11

"May the LORD make the woman (Ruth the Moabitess) who is coming into your home like Rachel and Leah, both of whom built the house of Israel; and may you achieve wealth in Ephrathah and become famous in Bethlehem".

Clearly Jehovah answered the people's prayers and the lineage did indeed become famous in Bethlehem when to the virgin Mary (from the line of David, from the line of Boaz) "was born Jesus,...(the) Christ." (Matthew 1:16)

This representative genealogy spans almost 850 years from Perez (1885BC) to David (1040BC).

Perez (6557) is a son of Judah and Tamar (Genesis 38:29).  Perez's father Judah was the fourth of Jacob’s 12 sons (Genesis 35:23) and the fourth son born to Jacob by Leah, who, overjoyed with the thought of bearing Jacob another son, named him Judah, meaning “praise” (Genesis 29:35).  So if we go backward in this genealogical record to Jacob, it would take us back probably sometime before 2000BC.

In light of the known ancestry of Perez, we can trace David’s lineage to extend back through Judah (Genesis 49:8–12), Jacob (Genesis 28:10–17), and Isaac (Genesis 26:24) to Abraham (Genesis12:1–3), to whom Jehovah had promised "2 And I will make you a great nation, and I will bless you, and make your name great; and so you shall be a blessing  and I will bless those who bless you, and the one who curses you I will curse. And in you all the families of the earth shall be blessed." (Genesis 12:2–3). This promise was repeated (almost 25 years later) "As for Me, behold, My covenant is with you, and you shall be" (Genesis 17:4). The book of Ruth then in a sense represents a partial fulfillment of Jehovah's promises to Abraham, promises that would ultimately be fulfilled in the Messiah! Beloved, read God's love letter to you and by faith grasp the promises He makes to you. You may not see the immediate fulfillment of those promises, but rest assured God will keep His Word and one day you will see the fulfillment!

Even if there are some omissions in the genealogy, its authenticity is attested by the fact that it would be very unlikely for a Jewish scribe recording the ancestry of David to have included a Moabitess unless the evidence was sure!

The first five names cover the period from the time of the entry into Egypt (Perez, Ge 46:12) to the time of Moses (Nahshon, Ex 6:23; Nu 1:7), while the remaining five belong to the period of the early settlement in Canaan to the closing years of the judges.

BBE: And Hezron became the father of Ram, and Ram became the father of Amminadab;
KJV: And Hezron begat Ram, and Ram begat Amminadab,
Young's Literal: and Hezron begat Ram, and Ram begat Amminidab,

KJV: And Amminadab begat Nahshon, and Nahshon begat Salmon,
Young's Literal: and Amminidab begat Nahshon, and Nahshon begat Salmon,

Septuagint (LXX):   kai Aminadab egennesen (3SAAI) ton Naasson kai Naasson egennesen (3SAAI) ton Salman

AND TO AMMINADAB WAS BORN NAHSHON, AND TO NAHSHON SALMON:

The father-in-law of Aaron (Ex. 6:23), who does not appear in 1Ch2:10, but is cited in Matt. 1:4 and Luke 3:33. Some Hebrew manuscripts also include Admin between Ram and Amminadab in Lu3:33.

According to Mt1:5, Salmon married Rahab the converted harlot, sometime after the fall of Jericho, and therefore Boaz was their son.

Matthew's  gospel parallels the genealogy found in Ruth 4 and records that...

KJV: And Salmon begat Boaz, and Boaz begat Obed,
Young's Literal:  and Salmon begat Boaz, and Boaz begat Obed,

Septuagint (LXX):  kai Salman egennesen (3SAAI) ton Boos kai Boos egennesen (3SAAI) ton Obed 

AND TO SALMON WAS BORN BOAZ, AND TO BOAZ, OBED: (1 Chr 2:11 2:12 Mt 1:5; Lu 3:32)

The fact that the genealogy connects Boaz and Ruth’s son Obed with Boaz rather than Mahlon does not mean he failed to perpetuate Mahlon’s reputation. The son would have been eligible to inherit from both Mahlon and Boaz. The Israelites regarded him as the son of both men. Naturally he was Boaz’s son, but legally he was Boaz and Mahlon’s son as well as Elimelech’s descendent.

Matthew includes the names of the Gentile mothers, Rahab and Ruth, writing that...

KJV: And Obed begat Jesse, and Jesse begat David.
Young's Literal: and Obed begat Jesse, and Jesse begat David.

Septuagint (LXX):   kai Obed egennesen  (3SAAI) ton Iessai kai Iessai egennesen (3SAAI) ton Dauid 

Jesus Christ’s lineage, through Mary, is traced to David (Mt 1:1-16; cf. Ro 1:3; 2Ti 2:8; Rev 22:16).

The Moabites were not to enter the congregation of the Lord “even to the tenth generation” (Dt 23:3). How interesting that the book of Ruth closes with a ten-generation genealogy that climaxes with the name of David!

Kidner writes that...

"God’s hand is all over history. God works out His purpose, generation after generation. Limited as we are to one lifetime, each of us sees so little of what happens. A genealogy is a striking way of bringing before us the continuity of God’s purpose through the ages. The process of history is not haphazard. There is a purpose in it all. And the purpose is the purpose of God."

Naomi’s return to Bethlehem and the covenant of marriage between Ruth and Boaz, explains why when

"it came about... that a decree went out from Caesar Augustus, that a census be taken of all the inhabited earth" that Joseph had to go up "to the city of David, which is called Bethlehem, because he was of the house and family of David (who was in turn of the house and family of Boaz and Ruth) in order to register, along with Mary, who was engaged to him, and was with child." (Luke 2:1-5).

From a human perspective, Ruth and Boaz are the reason the Messiah was born in Bethlehem but from a divine perspective this even occurred when and where it occurred because God had providentially ordained the historical events to perfectly fulfill prophecy. History is truly His story, as Jehovah alludes to in Isaiah declaring

"I am God, and there is no other; I am God, and there is no one like Me, declaring the end from the beginning and from ancient times things which have not been done, saying, 'My purpose will be established, and I will accomplish all My good pleasure.'" (Isaiah 46:9-10)

As alluded to throughout these notes on Ruth, Jesus has been pictured through the book, by Boaz the kinsman-redeemer as summarized in the table below:
 

Before the foundation of the world, in eternity past, God had planned to bring Ruth and Boaz together in the little town of Bethlehem which would be the "entrance point" for  Jesus, our Kinsman-Redeemer. How mysterious and sweet the sounds of the precious hymn...