The trinity is the belief that Jesus is God, the Holy Spirit is also God, and of course, God the Father is God. There are not three Gods; rather, each member of the Trinity is the same God. The Nicene Creed states that the members of the Trinity (personae in Latin, hypostases in Greek) are of one substance, and thus all are fully God, but distinct from each other. Actually, the trinity is often understood in far more complex terms from here, and there are often many misunderstandings as to its nature.
Many religions or groups, such as Islam and Jehovah's Witnesses, are quite steadfastly opposed to the idea of the trinity. Islam in particular states in...some part of the Qu'ran or another.... that Christians (or at least Christianity is implied from what I remember) are commiting one of the gravest sins in the entire universe by claiming that "God is more than one" or something like that. Many Islamic apologetics sites attempt to claim that Christianity believes in 3 separate Gods, and go from there in seeking converts. Jehovah's witnesses, as I understand it, also view the trinity as a belief that is allowing Satan to possess the minds of Christians worldwide, which may correspond to their disbelief in Jesus's deity. They are supposedly often prepared to debate on the nature of the trinity in door-to-door encounters.
Generally, these and other groups fail to understand that the trinity is not polytheistic, rather, it is the worship of one God, (Insert corresponding Isaiah quote here) who happens to be God, Jesus, and the Holy spirit all at once. This can sometimes be difficult to understand for some people considering instances in the New Testament where Jesus exibited somewhat un-God-like Characteristics, (Insert garden of Gesimine quote here) (Insert "even the Son does not know the time" quote here) but is most Biblically resolved from understanding of the Dual Nature of Christ. The Bible says that Jesus is both a man and God, (Insert corresponding quotes for each characteristic here) not merely 50 percent of one and 50 percent of the other, but completely both. Because of this, Jesus could both know and yet not know things, or exibit Godly or human characteristics, and when God was putting himself in this form, he may have even set many human limits for himself to make himself a true human (such as limiting his complete knowledge until he returned to heaven, or the like).
Complicated viewpoint here
The Trinity often becomes a very complex piece of theology at times, due to the fact that the word actually is not even in the Bible, and thusly it's understanding must be gleaned from a full understanding of the text in its entirety. The King James Version has one Bible verse (Might want to cite this somewhere) that refers directly to God being the Son, the Holy Ghost, and God, but most modern translations supposedly have not located this verse in the earliest and most reliable manuscripts. After this....well...I don't know off-hand, like I said, it's a veeeery complicated piece of theology at times, so um....someone might want to expand this.
This article is not yet very informative, and would benefit from additional input from Wikipedia, Theopedia or another reliable source.