I adapted the following from an article on MIT GCF's Skeptics Anonymous webpage, which I co-authored with Susan Kern and Cynthia Lo:
Christians believe that the three persons of the Trinity are all one God. Deuteronomy 6:4 states, "Hear, O Israel! The Lord is our God, the Lord is one!" He is a super-person, so to speak, His nature being so much more complex than our own that we cannot describe Him as a single "person." The doctrine of the trinity is perhaps the most difficult and perplexing to explain, since we are trying to describe the nature of the infinite God, which finite human beings are incapable of comprehending.
The term trinity describes a relationship not of three gods, but of one God who is three persons. Trinity does not mean tritheism, that is, that there are three beings who together are God, but the word trinity is used in an effort to define the fullness of the Godhead both in terms of His unity and diversity. The term trinity is never used explicitly in Scripture, but the concept is there from the beginning and specific passages such as Matthew 28:19, "baptizing them in the name [singular] of the Father and the Son and the Holy Spirit", refer explicitly to there being three "persons". All three persons of the trinity make an appearance at Jesus's baptism, as recorded in Mark 1:10-11, "As Jesus was coming up out of the water, he saw heaven being torn open and the Spirit descending on him like a dove. And a voice came from heaven: 'You are my Son, whom I love; with you I am well pleased.'" The "he" who saw this may be either Jesus or John the Baptist, who later testified about this event (John 1:32-34).
The church has rejected from the beginning heresies of modalism and tritheism. Modalism is the denial of the distinction of persons within the Godhead, claiming that the Father, the Son and the Holy Spirit are simply three "modes" of God expressing Himself. Tritheism reaches to the other extreme, that of falsely declaring that there are three beings who together make up God. The term "person" does not mean a distinction in essence, but a difference in subsistence. Jesus is different in subsistence from the Father or the Holy Spirit, but he is the same essence in terms of being. The Christian definition of God asserts that the three persons of the Godhead share the same essence, the same co-eternal existence, and the same will, but not the same mind, the same position, the same role, or the same relationship. All the persons in the Godhead have all the attributes of deity.
The trinity does not refer to "parts" of God and, unfortunately, human analogies fall short. An interesting but imperfect analogy may be found in ourselves, however. Human beings are composite creatures. Physically, we are trillions of cells working together to form the body, billions of neurons firing simultaneously to produce thought, two distinct hemispheres of the brain which "think" in different ways. Psychologically, we are a mess of conflicting emotions and ideas, each vying for primacy in our psyche. Spiritually, we are creatures of both soul and body, an uncomfortable mix filled with the strife between the physical and spiritual parts of our nature. Ultimately, one human person has less internal unity than the three persons of the trinity. And yet we never think of ourselves as more than one being.
The following is a traditional explanation for the roles of the three persons of God, taken from C. S. Lewis's Mere Christianity:
God is a Being which contains three Persons while remaining one Being, just as a cube contains six squares while remaining one body. But as soon as I begin trying to explain how these Persons are connected, I have to use words which make it sound as if one of them was there before the others. The First Person is called the Father and the Second the Son. We say that the First begets or produces the Second; we call it begetting, not making, because what he produces is of the same kind as Himself. In that way the word Father is the only one to use. But unfortunately it suggests that He is there first--just as a human father exists before his son. But that is not so. There is no before and after about it... The Son exists because the Father exists: but there never was a time before the Father produced the Son.
We must think of the Son always, so to speak, streaming forth from the Father, like light from a lamp, or heat from a fire, or thoughts from a mind. He is the self-expression of the Father--what the Father has to say. And there never was a time when He was not saying it... All these pictures of light or heat are making it sound as if the Father and the Son are two things instead of two Persons. So that, after all, the New Testament picture of a Father and a Son turns out to be much more accurate than anything we try to substitute for it... Naturally God knows how to describe Himself much better than we know how to describe Him. He knows that Father and Son is more like the relation between the First and Second Persons than anything else we can think of. Much the most important thing to know is that it is a relation of love. The Father delights in His Son; the Son looks up to His Father...
The union between the Father and Son is such a live concrete thing that this union itself is a Person. I know this is almost inconceivable but look at it thus. You know that among human beings, when they get together in a family, or a club, or a trade union, people talk about the "spirit" of that family, or club, or trade union. They talk about its "spirit" because the individual members, when they are together, do really develop particular ways of talking and behaving, which they would not have if they were apart. It is as if a sort of communal personality came into existence. Of course, it is not a real person: it is only rather like a person. But that is just one of the difference between God and us. What grows out of the joint life of the Father and Son is a real Person, is in fact the Third of the three Persons who are God.
This third Person is called, in technical language, the Holy Ghost or the "spirit" of God. Do not be worried or surprised if you find it (or Him) rather vaguer or more shadowy in your mind than the other two... Perhaps some people might find it easier to begin with the third Person and work backward. God is love, and that love works through men--especially through the whole community of Christians. But this spirit of love is, from all eternity, a love going on between the Father and the Son.
This explanation helps to illustrate a number of things. For one, the term "Word" applied to the Son in John 1 begins to make sense when we consider the Son as the "self-expression of the Father." Perhaps more importantly, it illustrates what is meant by 1 John 4:8, which declares that "God is love." We tend to minimize this, saying it means that God is loving. But throughout the Bible, the refrain is that God loves us because His very nature is love, and it would be unlike Him not to love us. But before humans and angels, what was there to love? What besides God is eternal? Love requires an object; the word is meaningless otherwise. Love could not be part of His eternal nature if He has not had some eternal object for His love. Instead, it would be something God learned to do once He had created someone to love. Only the trinity offers an explanation of how love can be a facet of the eternal nature of God, since contained in the three persons of the trinity are the subject, object, and expression of love. The three persons of the trinity are defined primarily by the relationship shared among them.
Update: (5/1/2004) I changed the phrasing to make it clear that Susan, Cynthia, and I are responsible for the article on Skeptic's Anonymous, not the whole page. Although... Cynthia as webmaster really is co-author on all of them, and I had a hand in quite a few. Susan may have joined in the debates on some of the other questions, but I don't really remember.