The free will problem is a metaphysical problem in Philosophy that arises when two deeply-held views come into conflict: the view that every event can be explained in terms of antecedent events and conditions, and the view that human beings have free will to decide to act in a certain way.
There are a great many schools of thought about whether human beings possess the ability to choose one course of action freely over of a number of alternatives. Our answer to this question will have a bearing on positions in areas such as moral responsibility, Philosophy of Science and Philosophy of Religion. The free will problem is most often motivated by a point of view called 'incompatibilism', which is itself motivated by the metaphysical doctrine known as 'determinism'.
(Causal) determinism is the view that all courses of events are caused, in turn, by other events and conditions. No course of events could have taken place other than the one that did in fact take place, given all the antecedent events and conditions. For instance, my dropping a glass from a great height on to a stone floor results in its smashing; this outcome was determined by the glass dropping in the first place, the fact that the floor was hard, the physical structure of the glass, the natural laws of gravity, and so on (and on and on and on). This kind of thinking goes into a lot of our day-to-day reasoning, not to mention the conclusions we draw in the scientific disciplines.
If everything that happens is determined in such a way, as seems almost undeniable as a premise (try thinking of any event that was not determined by some other set of events and conditions), then this is at odds with the idea that we have free will: when I make a choice, it will turn out that I could not have chosen otherwise, because events and conditions, physical or mental, determined that this choice was made by me. This leads us to the aforementioned position of 'incompatibilism'. Roughly, 'incompatibilism' is the doctrine that, if everything that happens happens because it was determined by some prior events and conditions, then, in the same way, all of my choices (which I thought were free choices) must also have merely been determined by some prior events and conditions. This is the free will problem: we feel as though we are making free choices when we choose to act in a certain way, but, as the incompatibilist points out, it is difficult to square this with the intuition that every event is metaphysically determined.
For the incompatibilist, (causal) determinism and personal autonomy are incompatible: when I choose, it will turn out that I could not have chosen otherwise, despite feeling as though I was making a free choice. Some notable counter-arguments to the incompatibilst position that we might appeal to include various 'compatibilist' models, or a Kantian conception of freedom. This latter approach was adopted by John-Paul Sartre in his famous defence of and response to freedom or free will as a central tenet of what it means to be a human being.