The problem of other minds is the problem of determining how we know that there are minds associated with the bodies we see walking around among us. This is, however, just the main problem. The problem can more accurately be expressed by breaking it into several steps.
1) We know that when you look at a red object you have the same perception as red that I do (i.e. We cannot know that if i was to inhereit your state of mind at that time, I would not call the sensation green)
1a) Therefore, we cannot know that other people have any sensations at all as opposed to automatic nervous reaction.
2) We can never know anothers mental state directly
Conclusion: We can never know that there exists any other minds but our own
In response to this problem there have been two main areas of attack. The reductionist view point with the likes of McDowell have tried to tackle the first two propositions 1 and 1a, by putting forth certain modes of expression such as being in pain as privilliged and allowing us direct access to the others mind. Thus, although they would admit from the problem of pretence, that at no one time can we claim to have access to anothers mental state, they are not permenantly unavailable to us.