This is a term that cannot be found anywhere in the Bible. It is a theological term used to describe a teaching which is derived from several passages in the Bible in both the Old and New Testaments.
It really says: THREE-UNITY This term refers to the fact that God, who really is one (in essence) has made Himself known (revealed Himself) to us in three ways.

When you get to know a person, you can tell somebody what he is like. We learn what God is like in three Persons:

-God the Father

-God the Son

-God the Holy Spirit.

Eph 3:9
And to make all see what is the fellowship of the mystery, which from the beginning of the ages has been hidden in God who created all things through Jesus Christ.

Gen 1:26
And God said, Let US make man in OUR image, after OUR likeness...

Gen. 1:2
And the Spirit of God moved upon the face of the waters.

Although the Lord Jesus and the Holy Spirit are revealed to us in our final understanding in the New Testament, it is clear to us that all three Persons of the Godhead were present from before time began.

GOD IS ONE

Deut 6:4
Hear, O Israel: The Lord our God, the Lord is one!

This was the specific message that came to Israel in a world that worshipped many false gods and idols and does not conflict with the N.T. teaching of the Trinity.
God is not an absolute unity, but rather a compound unity. We read about man and wife who shall be one flesh (Gen 2:24) and how a whole nation can act as one man...

Judges 20:11
So all the men of Israel were gathered against the city, united together as one man.

The word ONE used in these verses is a different word as used to convey absolute oneness. Deut 6:4 uses the plural Name of God ELOHIM. This shows us that although God reveals Himself to us in three persons, He is inherently ONE SPIRIT, indivisible in component parts, numerically one, free from composition.

The New Testament teaches this truth with the same clarity:
James 2:19

You believe that there is One God. You do well. Even the demons believe that - and tremble!

In the New Testament, God revealed that He is not only one, but a family of persons - an eternal, inexhaustible, and dinamic triune family of Father, Son, and Holy Spirit, who are one in will and purpose, love and righteousness.

THE TRINITY DEFINED:

In order to clearly understand that the Godhead is a Trinity in a Unity, we must not stress either the threeness or the oneness to the point where we fall into Tritheism (believing in three Gods) or forgetting the distinction of Persons (Sebellianism - that Father, Son and Holy Spirit are merely three aspects of manifestations of God)

From before time began, before any creature existed, there was an eternal fellowship between the three distinct Persons of the Godhead - every One of Whom was God, and was conscious of the other Two. Therefore there could also be a manifestation of the existence of LOVE. Otherwise, one could reason, how could God be Love - since God doesnt change?

1 John 4:16
And we have known and believed the love that God has for us. God is love, and he who abides in love abides in God, and God in him.

GOD IS LOVE FROM BEFORE ETERNITY.

The three Persons operate in singleness of mind and purpose, so that we can say that They are truly one:

-The Father CREATES

-The Son REDEEMS

-The Holy Spirit SANCTIFIES

In each work we see all three PRESENT. The Son and Holy Spirit co-operate in the creative work. The Father and the Spirit sent the Son to redeem. The Father and Son co-operate in the sanctifying work of the Spirit.

For the work of man's redemption, the Son entered the world in a new way when in New Testament He took upon Himself human nature. He was given a new Name - Jesus. At Pentecost, the Holy Spirit entered the world in a new way as the Spirit of Christ, being embodied in the Church, which is made up of all individual believers. We see that:

-The Father testified of the Son (Matt 3:17)
-The Son testified of the Father (John 5:19)
-The Son testified of the Spirit (John 14:26]
-The Spirit testified of the Son (John 15:26)

THE WORD TRINITY:

Although the word trinity is not in the Bible, it was used as early as 181 AD (Greek:Trias) and by Tertullian in 220 AD (Latin: Trinitas). The trinitarian formula is mentioned in the Great Commission (Matt 28:19) and in the benediction of the apostle Paul's second Epistle in the Corinthians (2 Cor 13:14)

In the Christian theology, the term TRINITY means that there are three eternal distinctions in One Divine Essence (being). These eternal distinctions are known as:

-Father
-Son
-Holy Spirit

The doctrine of the trinity means that within the being and activity of one God, there are these three distinct persons.