What exactly does it mean that Jesus Christ is the son of God? What does it mean to a Christian that Christ is the son of God? On the one hand, obviously, the normal biological meaning of son does not apply. On the other hand, "we are all children of God", but Jesus seems to be more so. 

That God offered up His only son as a sacrifice for humanity's sins is often used as an argument to demonstrate God's love for humanity. This would imply that God loves Jesus more than His other children, that this sacrifice was particularly hard, indicating that Jesus has a filial relationship with God in a sense that we would understand. It implies that sending His son to his death was extremely painful to Him, more so than the deaths of His other children.

Now, these arguments seem to me to be a clear anthropomorphisation of God, Christians seem to be attributing human characteristics such as the love of a father --not metaphorically as when referring to humanity as God's children, but in a very literal way-- to God. This seems to clash with another central tenet of Christianity which states that God is beyond our understanding, that we cannot fathom His plan. If so, then any attribution of human emotion to Him would be wrong.   

So, my question is how do Christians interpret Christ being the "Son of God"? What exactly does that mean? I realize the answer will depend on the particular denomination of Christianity whose views are being expressed. I am particularly interested in an overview of the more popular churches such as the Catholic, Protestant and Orthodox but welcome answers explaining the views of any group as long as the group in question is clearly stated.



NOTES

I have read the posts below, but though related, none of them addresses the same question:


If Christ is considered the 'Son of God' then how is He a part of a Trinity?
Does the Bible have any proof that Jesus Christ is the Only-begotten Son of God?

This one was quite interesting, and the accepted answer states that His "sonship" is unique, one-of-a-kind, and distinct from all others, which brings us straight back to my question, how is it distinct?
"Jesus Christ God's Only Son Our Lord"

Again, very interesting, but it while it explains the contradiction inherent in Christ being the only Son of God while we are all His children, it does not explain in what sense Christ is a son of God.
I'll try to answer this from several different views - the main three (Catholic, Protestant, Orthodox) and also the LDS view, since the LDS view is significantly different from the other three, and deserves a distinct treatment.



The mainstream Christian view (Protestant, Orthodox, Catholic) is covered by Apologetics Press in this article:  What Does it Mean to Say Jesus is the "Son of God"?.  The answer, for most Christians, has its roots in the Jewish understanding of the terms.  Bear in mind that this refers to the title "Son of God", and doesn't negate the teaching that Jesus is God.

An excerpt from the article:


  The earliest Christians were Jews who were familiar with at least two
  distinct applications of the term “son of God.” In the first place,
  the term had a general application to all Israelites. When their
  ancestors were held in Egyptian bondage, Moses was sent to Pharaoh
  with these words: “Thus says the Lord: Israel is My son, My firstborn.
  So I say to you, let My son go that he may serve Me” (Exodus 4:22-23;
  see also Hosea 11:1). Through the years, Yahweh loved, protected,
  comforted, and chastened Israel, just as a loving parent would nurture
  and discipline children (Malachi 2:10; Isaiah 66:13; et al.).
  
  The second usage was more specific. Historically, the term had a royal
  connotation for many nations of the Ancient Near East. It was
  commonplace for Egyptian, Babylonian, Canaanite, and Roman rulers to
  be called “son of God” (Fossum, 1992, pp. 128-137). These kings even
  were deified and surrounded by legends about their miraculous
  births—often including stories of gods copulating with humans
  (Sanders, 1993, pp. 243-245). This royal connotation also was known in
  Israel, although they did not deify their kings (O’Collins, 1995, p.
  117).
  
  When the New Testament writers referred to Jesus as “Son of God,” they
  sometimes employed the term in ways that echoed these two common uses.
  After those who threatened the life of the child Jesus died, Joseph
  was given instructions in a dream to return from Egypt to his
  homeland. When Matthew reported this event, he said it fulfilled Hosea
  11:1: “When Israel was a child, I loved him, and out of Egypt I called
  my son” (see Matthew 2:15). In other words, Jesus was God’s Son as an
  Israelite, and in a real sense, the True Israelite.


The link above is a fair Protestant understanding.  The Catholic teaching is the same, and though I can't currently find a reference, i believe it is the same in Eastern Orthodoxy.

Also, in the mainstream Christian view, Jesus, in His human incarnation, was conceived supernaturally (without sex) in the womb of Mary, enabling her virgin birth.  This would certainly set Him apart as different from the rest of us.  In a sense, he is the Son of God because God was the one who caused Mary to conceive.  (I'm probably saying that badly).

As for God having a unique love toward Jesus, in Gill's commentary on Matthew 3:17


  in whom I am well pleased. Jehovah the Father took infinite delight
  and pleasure in him as his own Son, who lay in his bosom before all
  worlds; and was well pleased with him in his office relation, and
  capacity: he was both well pleased in him as his Son, and delighted in
  him as his servant, Isaiah 42:1 he was pleased with his assumption of
  human nature; with his whole obedience to the law; and with his
  bearing the penalty and curse of it, in the room and stead of his
  people: he was well pleased with and for his righteousness, sacrifice
  and atonement; whereby his law was fulfilled, and his justice
  satisfied. God is not only well pleased in, and with his Son, but with
  all his people, as considered in him; in him he loves them, takes
  delight in them, is pacified towards them, and graciously accepts of
  them. It would be almost unpardonable, not to take notice of the
  testimony here given to the doctrine of the Trinity; since a voice was
  heard from the "father" in heaven, bearing witness to "the Son" in
  human nature on earth, on whom "the Spirit" had descended and now
  abode. The ancients looked upon this as so clear and full a proof of
  this truth, that they were wont to say; Go to Jordan, and there learn
  the doctrine of the Trinity. Add to all this, that since this
  declaration was immediately upon the baptism of Christ, it shows that
  his Father highly approved of, and was well pleased with his
  submission to that ordinance; and which should be an encouraging
  motive to all believers to follow him in it.
  
  Also from the Jamieson-Fausset-Brown Bible Commentary
  
  
  And lo a voice from heaven, saying, This is—Mark and Luke give it in the direct form, "Thou art." (Mr 1:11; Lu 3:22).
  
  
  my beloved Son, in whom I am well pleased—The verb is put in the
  aorist to express absolute complacency, once and for ever felt towards
  Him. The English here, at least to modern ears, is scarcely strong
  enough. "I delight" comes the nearest, perhaps, to that ineffable
  complacency which is manifestly intended; and this is the rather to be
  preferred, as it would immediately carry the thoughts back to that
  august Messianic prophecy to which the voice from heaven plainly
  alluded (Isa 42:1),


The LDS view is a bit different - LDS doctrine includes the idea that God is actively procreating in Heaven, and produces "spirit children".  These spirit children choose the human lives they will lead, so all of us pre-existed in Heaven with our actual spiritual father.  In LDS theology, Jesus was the first, and chose to be perfect and to suffer as he did to redeem man, as was God's plan.  Lucifer was another child of God, who wanted to offer salvation by removing the penalties of the law, thus destroying our free agency.

See Pre-Mortal Life - Mormon Wiki