One of the classic problems of philosophy, specifically the philosophy of mind.
The legacy of the problem can be traced back to Rene Descartes, (see Descartes' argument for dualism). It is a tricky issue to say whether he created it or discovered it, an issue full of the contradictions inherent in language and thought. It is similarly tricky to describe what the mind body problem is about. In fact, it seems to evaporate when you really describe it, making it seem, like other things to be a non-problem.
"Descartes cleavage," as my philosophy professor describes it, is the idea (fact?) that mind and body are different and separate. The mind and the body are separate things, and yet we seem to have both. Descartes proposed naively that the two interacted through the pineal gland. Other explanations of just "where" the mind "is" and how it communicates with the body seem equally silly.
Many philosophical theories have, over the past few centuries, tried to deal with the "problem" "created" by Descartes. Spinoza proposed the double aspect theory, in which both mind and body are effected by God, causing them to be in synch. There is also, idealism (only mind exists), materialism (only body exists), mind/brain identity (the mind just is the brain, except that theorists' emphasis is usually on the "brain" side) and other bouts of theoretical contortionism too numerous to mention.
My own view is that this problem is simply one of perception. We cannot resolve the mind/body problem because it is not a problem -- mind and body are the same thing. They are in no way separate things. They do not communicate any more than a computer communicates with its wiring. I've become more and more convinced of this the more I study about the science of the human brainmind. But, unlike many people, I have become less and less convinced that science, as it stands now, can tell the whole story of consciousness. Ken Wilber's books finally convinced me. This quote from The spectrum of Consciousness, talking about dualism in general, should sum things up nicely:
"...it is almost as if man were given two pictures of his body, one taken from the back, and the other from the front. In trying to decide which of these views was "really real," man divided into two camps: the Frontists, who firmly believed that only the picture taken from the front was real, and the Backists, who steadfastly insisted just the opposite. The problem was a tricky one, for each camp had to devise a theory to explain the existence of the other, and so the Frontists had just as much trouble exlaining the existence of the back as the Backists had in explaining the existence of the front. To avoid the contradiction, the Frontists spent their time running away from their backs, while the Backists were just as ingenious in devising ways to run away from their fronts. Occasionally the two would cross paths, yell obscenities at one another, and this was called philosophy."
Elsewhere, Wilber gets into the idea of a bodymind, a deeper reality of mind in which we acknowledge that mind and body are "not two." Try asking yourself: Would I say that I have a body, or that I am a body? If your answer was the first one, try defending the proposition in a debate. You may discover some interesting things.
After I noded this, I talked to a friend about it.
Me: I just noded the mind/body problem.
Him: What problem?
M: Exactly.
H: But what is it?
M: {insert explanation here}... but philosophers still treat it as a problem.
H: They must be bored.