The Trinity is never mentioned by that name in the Bible. While God, Jesus, and the Holy Spirit are all mentioned separately throughout the New Testament, there is only one passage which provides support for Trinitarian doctrine: 1 John 5:7-8 , which scholars call the Johannine Comma:
7 For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.
8 And there are three that bear witness in earth, the Spirit, and the water, and the blood: and these three agree in one.
This passage, the only Biblical support for the Trinitarian doctrine, is not found in the oldest and best Greek manuscripts of the New Testament, which instead read "These are three that bear witness: the Spirit, the water, and the blood, and these three are one." Scholars believe the Johannine Comma to be a later addition to the New Testament, inserted to justify the doctrines of the orthodoxy.
John 1:1 is often cited as in support of the trinity as well:
1 In the beginning was the Word, and the Word was with God, and the Word was God.
In this passage, there is no definite article before the word theos (god) in the third part of the sentence. Coptic translators and Origen of Alexandria (both c.200) indicated their belief that John included a definite article before theos when referring to the creator god, and left it out when he was not. The translation of John 1:1c as "the Word was God", therefore, is a topic of scholarly debate, as it could justifiably be translated "the Word was a god" without the reference vaguely supporting the Trinity doctrine.
Origen, however, noted that John was not consistent in his use of a definite article when writing in about god. Whether John meant the creator God in certain passages was a topic of debate as early as the second century, as evidenced by Origen's exegesis and the Sahidic Coptic translation being markedly anti-Gnostic. So it could be argued that both translations are equally valid.
Neither view, of course, addresses the possibility that John may well have been influenced by Greek polytheism, not consulted the other Gospel writers to make sure the details of his writings matched theirs, written a passage or two supporting a trinity, and been a bit inconsistent because he was just making it all up.
There have been a wide variety of different interpretations of the nature of the Trinity over the centuries. Modalism is an early Christian view that suggests that the three entities are different forms of a single God, in much the same way that water has solid, liquid, and gaseous forms. While some churches retain a modalistic interpretation, orthodox Trinitarians consider it to be heresy. Also considered heretical is Adoptionism, which posits that Jesus was a normal human who became divine, either at his baptism or his ascension. Unitarians deny the Trinity altogether, believing that there is one God, and that Jesus was merely a human prophet or perhaps a supernatural entity in his own right, but was not God in the flesh. Arianism was an early Christian doctrine which claimed that Jesus was created by God the Father, who later worked through Jesus to create the Holy Spirit, setting up a hierarchical godhead of separate entities. The Arian Controversy was a major reason for the Council of Nicea, which defined the Christian orthodoxy and effectively declared Arianism heretical. The Mormons believe that God, Jesus, and the Holy Spirit are separate entities with separate bodies, united in single purpose; this view is criticized by the orthodoxy as being a form of Tritheism, or worship of three separate gods. Jehovah's Witnesses see Jehovah as god, Jesus as an angel and the Holy Spirit as Jehovah's "active force" and not as an entity.