Today is Trinity Sunday, or the Solemnity of the Most Holy Trinity if you're not into the whole brevity thing. The doctrine of the Trinity is the greatest and most central mystery of our faith--you can't build up anything else if you can't answer the question, "Who is God?" Of course, answering this question is a bit trickier than answering "Who is Richard Nixon?" or "Who is this person who keeps calling me and hanging up?" The biblical and traditional data we have presents us with quite the puzzle:
We know that God is one. There is only one God. We know that.
We know that the Father is God. Obvi. No one disputes that.
We know that Jesus is God. God became man in Jesus Christ. Got it.
We know that the Spirit of God is God. God is the Spirit. Check.
But, wait a tic... doesn't that give us three gods? No, as St. Athanasius affirms for us: "The Father is God, the Son is God, the Spirit is God. Yet there are not three gods, but one God." And as the early centuries went on with their disputations and divisions, the ecumenical councils decreed that this indeed is the belief of the Church? But one could not be blamed for looking at the above and asking, "Are you sure there, St. A?" It's not easy to wrap your head around.
Many have wrestled with this and tried to formulate a theory that accounts for all the data while not sounding so contradictory-y. And many have failed. Which is good, on the hand, since it helps us to narrow things down by saying, "No, that's not it.... No, not that either.... Hmm, well... no."
I can't think off the top of my head of anyone who tried to say that God the Father isn't God. He's always been the sort of baseline, starting point, "Well, we can say at least this much." So we've got that goin' for us. Some denied the divinity of Christ, such as the Arians, who said he was almost, nearly, but-not-quite good, and super-swell creature but a creature all the same. Some denied the divinity of the Spirit, like the people whose names escape me right now, but who said that the Spirit was merely the force or power or energy of God. Some said that the one God was really truly only one but merely manifested Himself in different "modes," Father Mode, Son Mode, and Spirit Mode. (For you video gamers out there, this really complicates the secret "God Mode" status in some games.) These, and many other positions, were tried and tested and found insufficient by the early Church.
Basically, the false conceptions of the Trinity tend to waver between this modalism (God is really one, but looks like three) and tritheism (God is really three, but looks like one). And even most of the "helpful" explanations you hear in Sunday school or from the pulpit tend toward these: God is like an egg, with its shell, whites, and yolk (tritheism); like a shamrock, with its stem and three leaves (tritheism); like water, which can be liquid, ice, or steam (modalism). All analogies limp, as they say. They go as far as they go, but they're never perfect. These can be helpful in some ways, but they don't quite do the trick.
The orthodox doctrine is that there is one God, one divine nature, in which there are three Persons, the Father, the Son, and the Holy Spirit, all co-equal, all co-eternal, each as much God as the others. These three Persons are not three distinct substances or thingies, but they are one in every way apart from their relation to each other: The Father is not the Son nor the Spirit, the Son is not the Father nor the Spirit, the Spirit is not the Father nor the Son. The Father eternally begets the Son, the Father and Spirit eternally generate the Spirit, yet not in any sort of linear, sequential, "Oh, so the Father is really God, and then He produces the Son," etc., but rather in the way that as soon as a flame is lit, you already have with it its light and its heat, generated from the flame but inseparable from it. (Thank you, St. Hildegaard of Bingen... even though that one has its problems, too. Analogies limp.)
For a fantastic exposition of the doctrine of the Trinity, check out this excerpt from Frank Sheed's classic "Theology for Beginners."
Glory to the Father, and to the Son, and to the Holy Spirit, as it was in the beginning, both now and forever and unto the ages of ages. Amen.