What is the doctrine of the Trinity as defined by Christianity?
Obviously there are three entities involved: God the Father, Christ, and the Holy Spirit, but is there more to the definition?
As you already understand, the doctrine of the Trinity states that there is one God, who exists in three persons. The Father, the Son, and the Holy Spirit.
I'd like to expand on that, but there's not much I could expand on that hasn't already been covered in depth on this site. And while it's really not a complex doctrine, all sorts of heretical teachings come about based on a misunderstanding of it, or a rejection of it. Rather than going into any more detail here, I'd just recommend browsing the various questions tagged "Trinity" here.
On birth of Jesus Christ and His life, Death and Resurrection, it became essential to conceive the Doctrine of Trinity. Most obvious and direct evidence for Trinity originate from Bible depicting Jesus Christ and Holy Spirit with Nature of God. Therefore, if Father is God, Son is God and Holy Spirit is God than that’s it.
Father being God is acceptable to many which is as easy as accepting existence of God.
Jesus as God:
.2. He was crucified precisely for this charge when He declared that He is the Son of God thus making Himself equal with God.
.3. Think of the story of Jesus walking in the water, found in Matthew 14:22-33 and Mark 6:45-52. Most English translations hide the Greek by quoting Jesus as saying, "Fear not, it is I." Actually, the Greek literally says, "Fear not, I am." Those last two words are identical to what Jesus said in John 8:58, when he took upon himself the divine name "I am," which is the way God revealed himself to Moses in the burning bush in Exodus 3:14. So Jesus is revealing himself as one who has the same divine power over nature as Yahweh, the God of the Old Testament.
.4. One of the most striking is his forgiving of sin. If you do something against me, I have the right to forgive you. However, if you do something against me and somebody else comes along and says, "I forgive you," what kind of cheek is that? The only person who can say that sort of thing is God himself, because sin, even if it is against other people, is first and foremost a defiance of God and his laws. So along comes Jesus and he says to sinners, "I forgive you." The Jews immediately recognize the blasphemy of this. They react by saying, "Who can forgive sins but God alone?" Not only did Jesus forgive sin, but he asserted that he himself was without sin. And certainly sinless-ness is an attribute of deity.
.5. The way God was going to save the world was by his Son dying. God, in his divine nature does not die. So He had to come as a human being to accomplish that task. And Jesus believed he was the one to do it. Jesus said in Mark 10:45, "I did not come to be served but to serve and give my life as a ransom in place of many." This is either the highest megalomania or it's the example of somebody who really believes, as he said, "I and the Father are one." In other words, "I have authority to speak for the Father; I have power to act for the Father; If you reject me, you've rejected the Father."
Holy spirit as God:
1 Corinthians 2:11; 1 Corinthians 6:11; 1 Corinthians 6:19; 1 Corinthians 12:13; 2 Corinthians 3:5; 2 Corinthians 3:17-18; 2 Corinthians 13:13/14;Galatians 6:8; Ephesians 2:17-18; 1 Thessalonians 1:2-5; Hebrew 3:7-11; 2 Peter 1:21 etc.
The doctrine of the Trinity is the logical construct whereby three of the most obvious but otherwise seemingly contradictory bits of Scripture are reconciled. Namely:
The basic definition of the Trinity was popularized and credalized in the Nicene Creed, in which monotheistic Christians declare belief in God the Father, God the Son, and God the Holy Spirit. Furthermore, the Nicene Creed says that God the Son is "of one Being with the Father," and that the Holy Spirit proceeds "from the Father through (or and) the Son." This is the most widely held formulation of the Trinity - and is accepted by all but the most fringe of Christians.
Reconciling that God is one, and yet exists as three distinct persons is the central "problem" of the Trinity, and most heresies about the nature of God come from over stressing either their oneness or their distinctness.
It is always contentious. and other questions show why many analogies just don't work.
That this "mystery" has no earthly equivalent is not, however, an impossible defect - for it is simply understood that "God's ways are higher than our ways, and his thoughts higher than our thoughts." In other words, there is no law that says mortals need to be able to understand God's nature, just because there is no terrestrial equivalent.