The Trinity is a term for God, who is eternally existent in three co-equal and co-eternal persons, the Father, Son, and Holy Spirit. The Trinity is expressed in the Bible, in the Nicene Creed, and in the Sign of the Cross.
While generally viewed as a New Testament doctrine, there are a few references in the Old Testament that suggest the Trinity. The best known is in Genesis, when God uses the pronoun "us" to refer to Himself: "Let us make man in our image, in our likeness" (Genesis 1:26).
One of the major New Testament accounts containing the Trinity is the baptism of Christ by John the Baptist, when the Holy Spirit appears as a dove and the voice of God the Father says "Behold, this is My beloved Son, in whom I am well pleased." This is important because it shows all three persons of the Trinity present at the same time.
Trinity, or a belief in a Trinitarian God, is primarily a Christian idea. The New Testament clearly identifies three distinct persons or entities which would have to be God or closely associated with God. Trinitarianism has been the norm for Christian belief at least since the Nicene creed was formulated. Nevertheless some groups which consider themselves Christian interpret the three persons or aspects of God in different ways.
Tritheism: A belief that there are three distinct Gods, generally with the idea that they act in total agreement with each other. Judaism and Islam frequently assume that most Christians are guilty of this belief. which is totally inconsistent with normal monotheistic understanding of the Old Testament.
Modalism: A belief that there is only one real person or entity who is God, and that the different "personalities", Father, Son, and Holy Spirit, represent different aspects or modes of God.
Non-Christian trinities: Hinduism appears to have what might be considered a trinity. There is a creator, a maintainer, and a destroyer, but the three are said to be one in a way that might be considered a trinity.