is, so to speak, begotten of the source, and of one substance with it, and yet distinguishable from it; being that through which its glory is made manifest, and through which it enlightens all things. The Person of the Son is thus represented, not as of one apart from God, irradiated by His glory, but as Himself the sheen of his glory." (The Pulpit Commentary: New Testament; Old Testament; Ages Software  or Logos)

Wuest adds that

The word apaugasma is not preceded by the definite article, which fact makes the term highly descriptive of character or nature." (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Eerdmans or Logos)

Expositor's comments on radiance that

"In the Arian controversy (Ed note: Arius taught that the Son was a created being, inferior to God the father in nature and dignity though the first and noblest of all created beings) this designation of the Son was appealed to as proving that He is eternally generated and exists not by an act of the Father’s will but essentially… As the sun cannot exist or a lamp burn without radiating light, so God is essentially Father and Son.” (Gaebelein, F, Editor: Expositor's Bible Commentary 6-Volume New Testament. Zondervan Publishing)

the appearance of the glory (LXX = doxa) of Jehovah was like a consuming fire on the mountain top" (Ex 24:17)

It was there that God showed them

"His glory (LXX = doxa) and His greatness" (Dt 5:24).

Moses records that upon completion of the tabernacle,

the cloud covered the tent of meeting, and the glory (LXX = doxa) of the LORD filled the tabernacle." (Ex 40:34 see also Nu 14:10, 16:19 16:42).

The prophet Ezekiel described the departure (Ezekiel 10:4) and then foretold of the future return (Ezek 43:4 43:5) of the glory (doxa) of Jehovah from the Temple in Jerusalem.

In Exodus 33 Moses asks God

"I pray Thee show me Thy glory (LXX = doxa)!" And He said, "I Myself will make all My goodness pass before you...but..."You cannot see My face, for no man can see Me and live!" Then the LORD said, "Behold, there is a place by Me, and you shall stand there on the rock and it will come about, while My glory is passing by, that I will put you in the cleft of the rock and cover you with My hand until I have passed by. Then I will take My hand away and you shall see My back, but My face shall not be seen." (Ex 33:18 19 20 21 22 23) 

Jewish readers who would be very familiar with these OT descriptions of the glory (doxa) of Jehovah would have a clear understanding of the intent of the author's description of Jesus as the radiance of His glory but unlike Moses could look at the glorious face of God for

"the glory of God" is "in the face of Christ." (2Cor 4:6)

Jesus gives a correct opinion of all that God is, so that to see Jesus is to see God, for in Jesus Himself is the out shining of the majesty of the Father. The Son, being one with the Father (Jn 10:30), is in Himself, and ever was, the shining forth of the glory, manifesting in Himself all that God is and does.

Jesus' Own testimony was that

"he who beholds Me beholds the One who sent Me." (Jn 12:45)

Phillip queried Jesus

show us the Father and it is enough for us (Jn14:8)

Jesus replied that 

He who has seen Me has seen the Father" (Jn14:9) Explaining that "the Father is in Me" and "the words that I say to you I do not speak on My own initiative, but the Father abiding in Me does His works." (Jn 14:10)

John gives us the marvelous description of Jesus as

"the Word (Who) became flesh, and dwelt among us, and we beheld His glory, glory as of the only begotten from the Father, full of grace and truth" (Jn1:14).

John went on to say that

No man has seen God at any time; the only begotten God, who is in the bosom of the Father, He has explained Him." (Jn 1:18) (Comment: The Greek word for explained is exegeomai [ek = out  + hegéomai = tell, lead forward] means that Jesus as the God Man leads out or provides detailed information in a systematic manner of the glory of the Father. This word exegeomai  gives us our English word "exegesis" which describes the exposition of God's word, unfolding the interpretation through teaching.  Jesus then is the preeminent "Expositor" declaring the Father's glory thoroughly and particularly. "O come let us adore Him" "fixing our eyes on Jesus" (see note Hebrews 12:2)

Want to know what God says? Listen to Jesus. Want to know what God does? Watch Jesus. He is God incarnate in man.

Let's give an analogy realizing that it will be imperfect and can even distort the truth of the radiance of His glory if pressed too far. Jesus relates to God the way the rays of sunlight relate to the sun.  There is no time that the sun exists without the beams of radiance. They cannot be separated. If you put a solar-activated calculator in the sunlight, numbers appear on the face of the calculator. These are energized by the sun's radiance, but they are not what the sun is. The rays of the sun however are an extension of the sun. We see the sun by means of seeing the rays of the sun. So too we see God the Father by seeing Jesus for they are one God.

AND THE EXACT REPRESENTATION OF HIS NATURE: kai charakter tes hupostaseos autou: (2Co 4:4; Col 1:15 16)

flawless expression of the nature of God (Phillips)

the express image of His person (KJV, NKJV)

the very image of His substance (ASV)

the impress of His subsistence (YLT)

[the] exact expression of His essence (ALT)

the true image of his substance (BBE)

He is an exact copy of God's nature. (ICB)

the exact reproduction of His essence (Wuest)

all that God’s Son is and does marks him as God (TLB)

the very imprint of his being (NAB)

the exact imprint of God’s very being (NRSV)

everything about Him represents God exactly (NLT)

He is the perfect imprint and very image of [God’s] nature (Amp)

stamped with God's Own character (Moffatt)

the very image (impress) of His substance (Vincent)

the very representation of the divine essence (Vine)

Exact representation (5481) (charakter from charasso = to engrave and source of our English word character which describes one of the attributes or features that make up and distinguish an individual) was used in classical Greek of an engraver who mints coins or an engraving tool, a die, a stamp, a branding iron, a mark engraved, an impress or a stamp on coins and seals. Later it came to mean the impression itself, usually engraved, cut in, or stamped on in the form of a character, a letter, a mark or a sign. This impression or mark with its particular features was considered to be the exact representation of the object whose image it bore.

Charakter is a die made by an impress, like on a signet ring, the impression being identical although they are two separate entities. As a figure of speech charakter described a distinctive mark "impressed" on a person, by which he is distinguished from others. It is thus a characteristic of that person and was a Greek idiom for a person’s features. The author is saying that whatever the Divine essence is, Jesus is the perfect expression and thus affirms the deity of Jesus Christ and alludes to the the plurality of God. Jesus is distinct from God the Father and yet identical with Him.  Charakter conveys the idea of exact correspondence as when Jesus said that

"He who has seen Me has seen the Father" (Jn 14:9)

He who had seen Jesus had seen the Father because Jesus is the exact representation, the exact expression of the Father's attributes, nature, etc so that all that God is, Jesus is, and yet two distinct Persons of the Godhood. Jesus is all that God is, not has been given what He is! God hasn't given Jesus something. He already is! Paul concurs that Jesus "is the image of the invisible God" (see note Col 1:15)