The Trinity; the doctrine of the Trinity is a core Christian belief. Easy to say, impossible to unpack. So I'm not doing "research" on for this post, other than what I've done, probably resulting in a "stream of consciousness" sort of thing.
1. The Trinity is a mystery
Christians worship the Triune God; meaning that one God exists in three "persons.". Or three distinct persons are one God.
T.D.Jakes (who has been accused of "modalism," an old heresy) dislikes the word "person." You cannot see God as a "committee of three" - that implies that each member of the Trinity can be independent of the others, as if they are not truly One, but three that get together to be "God."
Yet "person" seems to be the best word that we have.
Each member of the Trinity is distinct, but none of them are EVER out of communion or unity with the others. Our God is ONE, yet three. Our God is THREE, yet one.
It is a mystery, and one that we may not fully understand even in heaven. Maybe the closest I can come (and still a really bad example) is the the three blind women oracles. They are three, but they function as one. They share an eye, and what one sees, they all see. They speak as one. They see as one, and they cannot function except as one. Yet the are three.
2. There is equality in the Trinity
No one in the Trinity is more equal than the others. The Trinity IS God. As one, God is complete, no person of the Trinity is more important, no person of the Trinity is more powerful, no person of the Trinity is more holy, no person of the Trinity is more eternal.
For a deeper look, read "The Forgotten Trinity" by James R. White
3. There is a hierarchy within the Trinity
This is hotly debated.
Many Complemetarians use the example of this hierarchy to demonstrate how absolute equals still live out a hierarchy. We look at the Trinity as an example of what life can look like. We believe the Trinity is a perfect example of persons who are equal, yet different in authority.
Many Egalitarians reject any notion of hierarchy because they believe it is impossible to ontologically live in submission (or in authority over) to an equal. In order to have equality, you must also have equality of authority.
But there is, in Scripture, as clear "sentness."
The Father sends the Son, who does not come in His own authority, but on the authority of the Father. "I have not spoken on my own authority; the Father who sent me has himself given me commandment what to say and what to speak. What I say, therefore, I say as the Father has bidden me..." (John 12:49,50).
The Son petitions the Father to send the Holy Spirit: "And I will ask the Father, and he will give you another Helper, to be with you forever," (John 14:16)
Jesus testifies to the hierarchy: "Truly, truly, I say to you, a servant is not greater than his master, nor is a messenger greater than the one who sent him." (John 13:16)
For more on the different understandings, read "The New Evangelical Subordinationism? : Perspectives on the Equality of God the Father and God the Son"
4. The Trinity is in "Oneness" - true unity.
Deuteronomy 6:4 “Hear, O Israel! The LORD is our God, the LORD is one!”
The "one" means a plurality that IS ONE.
5. There is purpose in the Trinity
God's two greatest gifts are creation and salvation - and all three members of the Trinity are involved in both these gifts.
Have we not all one Father? Has not one God created us? Why then are we faithless to one another, profaning the covenant of our fathers? Malachi 2:10
For by him all things were created, in heaven and on earth, visible and invisible, whether thrones or dominions or rulers or authorities—all things were created through him and for him Col. 1:16
The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters. Genesis 1:2
In salvation, all are present in Ephesians 1.
The Trinity is indeed a mystery. I cannot explain, but I know that God is not a modalist. The modern day heresy is known as "Oneness" theology.
The Trinity is a Biblical reality, as impossible to comprehend as eternity - all we can do is fall down and worship Him in His fullness, thanking Him for all good gifts.
For deeper reading: "The Forgotten Trinity" by James White