Every Saint Patrick's Day, I try to say something about the Godhead, the Holy Trinity. This isn't because I just know so much about it -- I don't -- but because St. Patrick, according to legend, taught the Irish about the Trinity using a shamrock. Which, as others have noted, is a fairly bad analogy.
The problem is that there simply are no good analogies for the Trinity. Practically all of them cause you to profess some heresy or another (partialism, modalism, Arianism, Sabellianism, and a host of other bad -isms). So we're left with explaining things by not really explaining things, instead leaving them wrapped in mystery. Our best attempt at an explanation comes from Athanasius, an Early Church Father famous for his commitment to the doctrine of the Trinity:
"And the catholic [universal] faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence. For there is one Person of the Father; another of the Son; and another of the Holy Ghost. But the Godhead of the Father, of the Son, and of the Holy Ghost, is all one; the Glory equal, the Majesty coeternal. Such as the Father is; such is the Son; and such is the Holy Ghost. The Father uncreated; the Son uncreated; and the Holy Ghost uncreated. The Father unlimited; the Son unlimited; and the Holy Ghost unlimited. The Father eternal; the Son eternal; and the Holy Ghost eternal. And yet they are not three eternals; but one eternal. As also there are not three uncreated; nor three infinites, but one uncreated; and one infinite. So likewise the Father is Almighty; the Son Almighty; and the Holy Ghost Almighty. And yet they are not three Almighties; but one Almighty. So the Father is God; the Son is God; and the Holy Ghost is God. And yet they are not three Gods; but one God. So likewise the Father is Lord; the Son Lord; and the Holy Ghost Lord. And yet not three Lords; but one Lord. For like as we are compelled by the Christian verity; to acknowledge every Person by himself to be God and Lord; So are we forbidden by the catholic religion; to say, There are three Gods, or three Lords. The Father is made of none; neither created, nor begotten. The Son is of the Father alone; not made, nor created; but begotten. The Holy Ghost is of the Father and of the Son; neither made, nor created, nor begotten; but proceeding. So there is one Father, not three Fathers; one Son, not three Sons; one Holy Ghost, not three Holy Ghosts. And in this Trinity none is before, or after another; none is greater, or less than another. But the whole three Persons are coeternal, and coequal. So that in all things, as aforesaid; the Unity in Trinity, and the Trinity in Unity, is to be worshipped. He therefore that will be saved, let him thus think of the Trinity."
In short, God the Father, God the Son, and God the Holy Spirit are all the same God while still being three distinct Persons. It's a bit hard (okay, pretty much impossible) to wrap our heads around, I know, but this is the mystery of God. If it gets too hard, we can always take comfort in another aspect of the nature of God: God is love, and He loves us whether we figure out everything or not.