The doctrine of the Trinity - in it's simplest form is the belief of Father, Son and Holy Spirit - yet, it is a bit more complexed.
The Trinitarian view is this. God is of ONE essential essence but in three distinct persons. This would be expressed mathematically as: 1x1x1=1 or put another way - The Father is God, the Son is God and the Holy Spirit is God - since each Person, is of one essence, they are one God. There are many mysteries in the Bible and surly, the Trinity is one. So too, the Incarnation of Jesus Christ. God, the Son, coming to earth in the flesh. The Divine nature and human nature residing within the same body.
While the word "Trinity" does not appear in the Bible, there are several good examples of this union in Scripture. Here are a few:
Gen_1:26 And God said, Let us make man in our image, after our likeness:
Note the use of the plural pronouns "us" and "our". This expression proves, that we have three Persons, taking counsel with one another. Some say God expresses Himself as three persons but God is not three persons. If this was true, then Gen. 1:26 - has God talking to Himself - since there are no other persons to talk too.
While there are three distinct Persons, within the Godhead - this does not mean... three differing thoughts or views. All three are of one mind and have the same thoughts. Remember... God is Eternal and as such, has never had a thought, nor will have a thought, that He has not already thought. God is unchanging. All three Persons of the Godhead, are of the same mind, on all things. The Father planned every aspect of Creation and Redemption and the Son and the Holy Spirit agreed. The Son subjected Himself to His Father and the Holy Spirit subjected Himself to both - Yet, all three are equal as God. (John 10:30; 17:11) Complicated indeed but not impossible to comprehend.
Luke 3:21 Now it came to pass, when all the people were baptized, that, Jesus also having been baptized, and praying, the heaven was opened,
Luke 3:22 and the Holy Spirit descended in a bodily form, as a dove, upon him, and a voice came out of heaven, Thou art my beloved Son: in thee I am well pleased.
In verse 22 - we see all three Persons involved in the baptism of Jesus. 1.) The Holy Spirit descended upon Jesus, 2.) The Son was in Jesus, for He is the God-Man, 3.) The Father spoke from heaven about His Son.
If God was, as the Unitarians believe, One God - manifesting Himself as three persons - this verse, singlehandedly, would make that idea impossible. If we followed the Unitarian view, we would have to change the last part of verse 22, to read: ".. Thou art Myself: In Myself I am well pleased.".
I hope that in some small way, this might have helped.