Humans are made up of the body, mind, and soul. It's easy for us to seperate the body from the other two. We've all seen a dead person before, or can at least imagine a dead person. That is the body, without the mind or soul present. The mind and soul are a bit harder seperate from each other, since they are both rather intangible. But if you think of a person who has suffered massive head trauma or fallen victim to a debilitating disease of the mind, the might still be alive, but their mind might have been destroyed. We assume their soul is with their body until the time of death, so then if the mind is gone, but the soul is sound, we must assume they are different entities. There is other evidence as well*.
Ok, so having pretty much established the existence of the body, mind, and soul, we can move on to how that translates to the Holy Trinity. Jesus came to earth in man-form, so obviously he would translate to the body. God the Father is the designer of the universe, the planner, and the voice that spoke everything into existence. This would most closely correlate to the mind. This leaves the Holy Spirit to correlate to the soul, and indeed that makes sense. The Holy Spirit is the Comforter that comes to live inside you when you accept Christ as your savior. He lives inside you alongside your soul.
Now obviously there is not a definite seperation between all these. If you get hit in your head hard enough, the mind suffers, despite the blow happening to your body. If your mental state is haywire it affects your body and your spirit. Your spiritual life has a profound effect on your state of mind. The Holy Trinity is very similar, and always connected, except when Jesus died for our sins. I think this is what had to happen for our sins to be forgiven. Jesus was seperated from the Father and the Spirit, because sin can not be in God's presence. Our sin seperated Jesus from His Father. Hence the "My God, my God! why have you forsaken me?". And as we see with humans, the body is the easiest to seperate. Jesus received a new body, as will we.
I hope this has helped clarify, if you were having trouble understanding. Obviously, this isn't meant to be all comprehensive, but just a general guide so people can begin to wrap their mind around the trinary nature of God.
*My other evidence comes from my experience counseling people. When people forgive themselves for things that they have done in the past, it is removed from the spirit, but the mind might still remember it. They can recall the event, but not feel the shame and spiritual seperation from God that they once did.