Some of questions asked in the philosophy of mind are simply what the mind is, what mental events are, what each type of cognitive process involves, and what consciousness is. Each of these involve some very hard questions.
The first question is: What is the mind? Is the mind nothing more than a series of particular thoughts, feelings, and so forth, or is it something over and above those particular thoughts, feelings, and so forth? When we ask that, we are simply asking what the relation is between the mind and mental events. In other words, we could simply restate the question in terms of mental events, like this: Is the mind nothing more than a series of mental events, or is it something over and above the mental events that we say occur "in" it? This isn't a question we are going to try to answer.
But of course there are other questions we could ask about what the mind is; we might raise the mind-body problem. Are our minds something that goes beyond our physical bodies? Suppose we think that the mind is a substance of some sort -- a MentalSubstance. We might still ask: Is there some way to explain what the mind, a mental substance, is in terms of physical substance? Or will we maintain that the mind is something totally different from physical bodies, and that we cannot explain what the one is in terms of the other at all?
Suppose instead that we deny that the mind is some mysterious substance, and we hold instead that there are only mental events and that "the mind" designates no more than a series of mental events? We can still inquire about the relation between mind and body a different way, in terms of the relation between mental events and physical events. We can ask: Are mental events totally different from physical events, so that you can't explain what mental events are in terms of physical events; or are mental events somehow explainable as being the same as physical events? For example, when John feels a pain, a mental event is occurring; now is that pain even possibly the same as something that occurs in John's brain, such as the firing of some special group of neurons?
So far we've presented several different questions that the philosophy of mind asks: What is the mind, a substance or just a series of mental events? Is the mind somehow reducible to, or explainable in terms of, the body? Are mental events somehow reducible to, or explainable in terms of, physical events? Each of these questions are ways of interpreting the more ambiguous questions we started with, such as, "What is the mind?" and "What are mental events?"
We can also ask questions about the different specific cognitive processes, and of course we might ask what cognitive processes in general are supposed to be. In that case, we'd be asking what distinguishes a cognitive process from any other kind of process. That is another way of putting the mind-body problem. We can also ask a series of more specialized questions, about each individual cognitive process.
Take perception as an example. Philosophers ask what is going on when we perceive something -- when we see, hear, taste, touch, and so on. But philosophers are not interested in the particular mechanisms that allow us to see -- for example, they do not study the shape of the eye or how the optical nerve carries information to the brain. They are interested in even more basic questions. They ask: Do we perceive physical objects directly with our senses, or do we form mental images of some sort, which we use to represent physical objects and their properties? These are questions raised by the philosophy of perception. The philosophy of perception is all about how our minds come in contact with the world outside our minds.
Another example is the will, or volition. When we choose to do something we are using our wills, or engaging in volition. There is, of course, one special and very difficult question that philosophers ask about this process, namely, is the will free? If Mary decides to walk across the room, that seems to be entirely up to her; she could have chosen otherwise. But if the universe is determined, and especially if our will really is after all just a physical process, then it certainly does seem as though Mary didn't have control over everything that led up to her deciding to walk across the room. So was she free or wasn't she? See also: Free will and determinism.
Finally, consider some questions about consciousness. What is it? We say that there is something it's like to "be watching a baby". When we look at a baby we are conscious of the baby. Is there some way to explain what makes a mental event, like looking at a baby, conscious? Well, what could we explain consciousness in terms of? If in terms of some physical process, then we face the same old mind-body question in yet another form: Can consciousness be reduced to, or explained in terms of, mere physical processes? Some people have said, vociferously, definitely not. How could a hunk of grey matter in your brain be the same as the awareness of a pain? Awareness is a totally different kind of thing from grey matter in your skull.
But that is only one question that can be asked about consciousness. There are other questions. Indeed there are a lot of questions that can be asked about all the other mental functions, such as memory, forming concepts, reasoning, the emotions, and so on.
Some people who work in the area of philosophy of mind are Daniel Dennett, Owen Flanagan, John Searle, Patricia Churchland[?], David Chalmers, Gerald Edelman, Francis Crick, Ned Block[?], Georges Rey, Thomas Nagel, Paul Churchland[?].