Okay, this isn't necessarily going to be accurate, it's just how I understand it. The Trinity is made up of three people, all of whom are God - God the Father, God the Son, and God the Holy Spirit. So, three people, but also one person. Not tremendously comprehensible, and basically I suspect that none of us really gets it, so please don't be surprised if you come across an explanation that completely contradicts mine - both explanations are probably wrong!
But anyway - I always figure that they're all God because they've all got God's personality. You know how identical twins start off being dead similar, but then diverge as time goes on? Well, the Bible says that God doesn't change - he's always been the same and always will be. So if God was three identical twins, they wouldn't just be similar, they'd be the same person - if you put any of the three in a particular situation, all their thoughts/feelings/etc. would be the same. Okay?
However, all three members of the Trinity are in different situations, and have different amounts of knowledge and different abilities. Jesus often prayed to his Father, which seems strange, given that they're the same person, only if you think about it, God the Father must have been able to think a lot more clearly than God the Son (Jesus), because Jesus was human, and therefore full of irrational feelings and temptations and stuff. So it's a bit like if I was, like, with my girlfriend, and really not feeling like behaving, and I could actually contact myself at another period of time when my mind was calm and clear, and ask myself what I should do. Does that make sense?
Anyway, as far as I can gather, God the Father is the guy who sees everything super clearly, and is sort of in H.Q. sitting at the control seat, God the Son is the same guy, but with a human body (although it's now a ressurected one, like the one we'll get after we die) and knows exactly what it's like to be human, and God the Holy Spirit is the same guy in a spiritual form on earth, who is everywhere, but especially lives in the hearts of Christians. So it's perfectly correct to say that I've got Jesus living in my heart, even though it's really the Holy Spirit, 'cause they're the same guy!
So that's how I understand the trinity - sorry if it didn't make sense! The thing I really like about it is that in Jesus we have an example of how God would behave as a human being, with no special powers or anything. It was just pure force of character, which impresses me more than anything else God has ever done. And it was blinkin' nice of him to do it too - brilliant guy, love 'im!