Key Christian belief: Jesus is God
Q: Does that mean Jesus is the same as God?
A: Yes and no. Jesus is God, but there is more to say about Jesus and God than just that.
1. There's more to say about God: For the Father is God (#1) and the Spirit is God (#3)
2. There's more to say about Jesus: for he is true man as well as true God.
Q: OK, so does that means Jesus is just part of God?
A: No, Jesus is God, not part of God.
For God has no parts.
The doctrine of the Trinity says the Father is God (#1), not part of God,
and the Son is God (#2), not part of God, etc.
That is why analogies for the Trinity like the three parts of an apple are just plain wrong.
The concept of Hypostasis.
To reinforce this point, Trinitarian doctrine needs a term that will indicate that Father, Son and Spirit are each a complete individual being, not a part of anything. That term is hypostasis. It is a very abstract term meaning, "complete individual being (of whatever kind)." E.g., my hand is not a hypostasis, but I am. My dog's ear is not a hypostasis, but my dog is.
Why do we need such an abstract term? Because we cannot use a term for some specific kind of being, such as dog or man or god. For three individuals of the doggy kind are three dogs, three individuals of the human kind are three humans, but the three divine hypostases are not three gods. The peculiar arithmetic of the doctrine of the Trinity, where one and one and one do not add up to three, makes it impossible to use specific kind terms, so we must use the most abstract term possible for "complete individual being (of whatever kind)."
The term Person
"Hypostasis" comes from Greek, "person" from Latin. For persona was the Latin term the Western church used to translate the Eastern church's Greek term, hypostasis. It is important to see that when the term persona was originally used in Christian doctrine, it did not mean "personality." It literally meant mask (from the Latin verb per-sonare, to sound through) but figuratively, it meant roles in a drama. For in ancient dramas, actors on stage wore masks. That's why, to this day, the cast of characters in a playbill is sometimes labeled dramatis personae, literally the masks of the drama.
So the term "person" is a reminder that in the drama of salvation narrated in Scripture, God appears as three different characters, three distinct individual beings: God in heaven, Christ on earth (who is then exalted to God's right hand) and the Holy Spirit given to the church. Each of these characters is a complete individual being who is fully God, not a mere mask or role. (In that sense, the ancient meaning of persona is be a bit misleading: the doctrine of the Trinity can't use any term of human language without bending it a little) . But neither does "person" simply have its modern meaning, as if it meant that God had three "personalities." Three different personalities would be three different Gods.
The concept of person or hypostasis is particularly important because it links the doctrines of Trinity and Incarnation. In the one doctrine, Christ is the second person of the Trinity. In the other, that same person is both God and man. The fact that the same person appears in both doctrines is the crucial link between the two doctrines, and indeed the crucial thing the church has to say about who Christ is: the same one (the same person or individual being) who is the eternal Son of God, begotten from the Father as the second person of the Trinity, is also God incarnate, true God and true man, born of the virgin Mary, crucified under Pontius Pilate, raised from the dead on the third day, and ascended in his human flesh to the right hand of God on high.