Who Are the Sons of God?
There are many Bible verses about the sons of God (or children of God), but they talk about different sons, so we first need to look at what a son of God is.
I will argue that a son of God is someone who reflects the character of God.
So God created man in his own image, in the image of God created he him; male and female created he them.
- Genesis 1:27 (KJV)
And what is this character?
In this the children of God are manifest, and the children of the devil: whosoever doeth not righteousness is not of God, neither he that loveth not his brother.
- 1 John 3:10 (KJV)
So, in other words, a son of God is righteous (obedient to God's law), and loves his brother.
When we obey God's law we reflect God's character (no sin).
There are three kinds of sons of God.
- Begotten
- Created
- Adopted
When God begats a Son, that Son is God. When God creates a son, that son is a created being (not God).
Jesus
Jesus is of course the only begotten Son of God (capital S).
For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.
- John 3:16 (KJV)
Because Jesus is a begotten Son, it means that He is God by definition. No one else can claim to be a Son of God, other than Jesus.
Let this mind be in you, which was also in Christ Jesus: Who, being in the form of God, thought it not robbery to be equal with God: But made himself of no reputation, and took upon him the form of a servant, and was made in the likeness of men: And being found in fashion as a man, he humbled himself, and became obedient unto death, even the death of the cross.
- Philippians 2:5-8 (KJV)
We are predestined to become like Jesus (no sin).
For whom he did foreknow, he also did predestinate to be conformed to the image of his Son, that he might be the firstborn among many brethren.
- Romans 8:29 (KJV)
Men
When Adam was created, he was a son of God (lowercase s).
Which was the son of Enos, which was the son of Seth, which was the son of Adam, which was the son of God.
- Luke 3:38 (KJV)
So at creation Adam was a son of God, but then came the Fall, and he became a son of sin (like us).
For all have sinned, and come short of the glory of God.
- Romans 3:23 (KJV)
The only way for us to become sons of God, is through Christ Jesus (repentance). This process is called adoption.
For as many as are led by the Spirit of God, they are the sons of God. For ye have not received the spirit of bondage again to fear; but ye have received the Spirit of adoption, whereby we cry, Abba, Father.
- Romans 8:14-15 (KJV)
But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name: Which were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God.
- John 1:12-13 (KJV)
For ye are all the children of God by faith in Christ Jesus.
- Galatians 3:26 (KJV)
It is very possible that even Adam and Eve after their fall were saved.
They [Adam and Eve] sincerely repented of their guilt, believed the promise of God, and were saved from utter ruin.
- Ellen White, Signs of the Times, April 22, 1886
Angels
What about the angels? Are they sons of God too? Do they reflect the character of God?
The Bible doesn't say either way, but yes, I do believe so. The angels do reflect the character of God (His law and commandments).
Bless the Lord, ye his angels, that excel in strength, that do his commandments, hearkening unto the voice of his word.
- Psalms 103:20 (KJV)
Ellen White, who is an inspired writer, says this.
A complete offering has been made; for "God so loved the world, that he gave his only-begotten Son,"-- not a son by creation, as were the angels, nor a son by adoption, as is the forgiven sinner, but a Son begotten in the express image of the Father's person, and in all the brightness of his majesty and glory, one equal with God in authority, dignity, and divine perfection. In him dwelt all the fullness of the Godhead bodily.
- Ellen White, Signs of the Times, May 30, 1895 par. 3
The closest Bible verse I could find to support this, is this one.
One God and Father of all, who is above all, and through all, and in you all.
- Ephesians 4:6 (KJV)
Some say that the verse below rules out that the angels are sons of God.
For unto which of the angels said he at any time, Thou art my Son, this day have I begotten thee? And again, I will be to him a Father, and he shall be to me a Son?
- Hebrews 1:5 (KJV)
But note that Son is written here with a capital S. That means it's referring to Jesus.
So if we look at the context of this verse (Jesus being compared to the angels), I think the verse should be read as:
For unto which of the angels said he at any time, Thou art my [begotten] Son, this day have I begotten thee? And again, I will be to him a Father, and he shall be to me a [begotten] Son?
- Hebrews 1:5 (KJV)
Angels are ministering spirits.
Are they not all ministering spirits, sent forth to minister for them who shall be heirs of salvation?
- Hebrews 1:14 (KJV)
They are servants.
Then saith he unto me, See thou do it not: for I am thy fellowservant, and of thy brethren the prophets, and of them which keep the sayings of this book: worship God.
- Revelation 22:9 (KJV)
The angels refer to Jesus as their Commander.
While it was yet dark, the watching angels knew that the time for the release of God's dear Son, their beloved Commander, had nearly come. As they were waiting with the deepest emotion the hour of His triumph, a mighty angel came flying swiftly from heaven.
- Ellen White, Early Writings, p. 181
So they are definitely a little bit different from us, e.g. they are spirits, we are flesh and blood, but they do keep the commandments, so they must have been created in God's image.
It's also interesting to see that the angels are either referred to as "angels of God", or "stars", but never really as "sons of God".
Also I say unto you, Whosoever shall confess me before men, him shall the Son of man also confess before the angels of God: But he that denieth me before men shall be denied before the angels of God.
- Luke 12:8-9 (KJV)
I believe there are two groups in the verse below.
When the morning stars sang together, and all the sons of God shouted for joy?
- Job 38:7 (KJV)
(This verse takes places shortly after the Creation)
A star means angel.
The mystery of the seven stars which thou sawest in my right hand, and the seven golden candlesticks. The seven stars are the angels of the seven churches: and the seven candlesticks which thou sawest are the seven churches.
- Revelation 1:20 (KJV)
What I'm trying to say is that in the Bible the sons of God never refer to the angels.
However, I do strongly believe that they are sons of God, but that maybe they have been given different titles (angels, stars, etc) to distinguish one group from the other.
Fallen
Someone who has fallen (sinned) is not considered a son of God (anymore). In their fallen state they are called sons of sin (or sons of the devil).
Ye are of your father the devil, and the lusts of your father ye will do. He was a murderer from the beginning, and abode not in the truth, because there is no truth in him. When he speaketh a lie, he speaketh of his own: for he is a liar, and the father of it.
- John 8:44 (KJV)
Fallen beings are never called sons of God.
Son of Perdition
Perdition = complete and utter ruin.
Judas is considered a son of perdition.
While I was with them in the world, I kept them in thy name: those that thou gavest me I have kept, and none of them is lost, but the son of perdition; that the scripture might be fulfilled.
- John 17:12 (KJV)
The pope (man of sin) is also considered a son of perdition.
Let no man deceive you by any means: for that day shall not come, except there come a falling away first, and that man of sin be revealed, the son of perdition.
- 2 Thessalonians 2:3 (KJV)
And I guess, the devil (and his angels) fall into this category as well.
Other Worlds
The phrase sons of God can sometimes also refer to the inhabitants of other unfallen worlds.
There is the throne, and around it the rainbow of promise. There are cherubim and seraphim. The commanders of the angel hosts, the sons of God, the representatives of the unfallen worlds, are assembled. The heavenly council before which Lucifer had accused God and His Son, the representatives of those sinless realms over which Satan had thought to establish his dominion,- all are there to welcome the Redeemer. They are eager to celebrate His triumph and to glorify their King.
- Ellen White, The Desire of Ages, p. 834
But that's probably more something for another day. 
Conclusion
We need to look closely at the context to understand what group is being referred to when it says sons of God. Especially around Genesis 6.
The best definition I have seen for sons of God, is this one.
For as many as are led by the Spirit of God, they are the sons of God.
- Romans 8:14 (KJV)
To me it means that no created being can keep God's law without the help of the Holy Spirit.
Either way, I think it's absolutely awesome to be called a child of God.
Beloved, now are we the sons of God, and it doth not yet appear what we shall be: but we know that, when he shall appear, we shall be like him; for we shall see him as he is.
- 1 John 3:2 (KJV)
What an honor. 