One of the most fascinating problems philosophers wrestle with is the problem of consciousness. What exactly is consciousness? How is it produced? What is its relationship to the brain? Etc.
Usually we think of consciousness as a state of being aware, but how does awareness arise from inanimate matter? How can atoms and molecules make us aware of ourselves and of the world external to ourselves? How does this all happen? How, for example, does an electro-chemical reaction in the brain generate the sensation of red?
Peter at Conscious Entities has an interesting discussion of what he calls the Three and a Half Problems of Consciousness.
The problems he discusses are the problems of Qualia, Intentionality, Morality, and Relevance. The article is a good primer on what is one of the most cutting edge fields in contemporary philosophy and also one of the most vexing because the problems seem so intractable.
Qualia are the sensations that we experience as part of our conscious awareness of the world. The sensations of color, sound, emotions, etc. are the qualia of our experience.
Intentionality refers to the fact that thoughts are about things. How is it that a particular flow of atoms, chemicals, and electricity can be about something.
The problem of moral responsibility concerns, among other things, the question of how we can be responsible for our actions unless we somehow cause them, but if we are bound by the laws of physics then in what sense is it us who causes our actions rather than the laws of physics which constrain us?
Anyway, read what Peter has to say on these things and you'll be well on your way to a graduate degree in the philosophy of mind.