Bible Teaches Doctrine of Trinity

What does it mean when we say that God exists in a trinity?

It is the teaching that there is one God in three persons: the Father, Son and Holy Spirit. Because this one God exists in three persons, we can say that the Father is God, the Son is God and the Holy Spirit is God. But since they are three persons, the Father is distinct from the Son who is distinct from the Holy Spirit. All three persons exist in equality in the trinity. No one is superior to the other two persons.

Where is the doctrine of trinity taught in the Bible?

The doctrine of trinity is not found in a single verse of the Bible. It is the result of putting together in a consistent way what the Bible teaches about God.

There is only one God.

(Deu 6:4)  Hear, O Israel: The LORD our God, the LORD is one.

The Father is God.

(Mal 2:10)  Have we not all one Father ? Did not one God create us?

Jesus is God

The Holy Spirit is God.

(Acts 5:3-4)  Then Peter said, "Ananias, how is it that Satan has so filled your heart that you have lied to the Holy Spirit and have kept for yourself some of the money you received for the land? {4} Didn't it belong to you before it was sold? And after it was sold, wasn't the money at your disposal? What made you think of doing such a thing? You have not lied to men but to God."

Putting it all together.

Because there is only one God, but there are three persons - Father, Jesus and the Holy Spirit - who are also God, this one God has to be a triune God that exists in three distinct persons.