Gah!
Where knowledge is defined as justified true belief, a priori knowledge is the set of true beliefs that are justified without appeal to sensory experience. In contradistinction, a posteriori knowledge is the set of true beliefs that depend on sensory experience for justification. In this view, all true beliefs must be justified to count as knowledge--the a priori/a posteriori distinction concerns the manner of such justification.
The true belief that every bachelor is an unmarried man can be justified without needing to round-up all bachelors to see if they're unmarried, because the fact just follows from the meaning of 'bachelor'. This is a classic example of a priori knowledge. However, a priori knowledge need not always be as trivial as this. For example, consider the true belief that no highest prime number exists. Few people find this obvious, but it follows as surely from the nature of numbers as being unmarried follows from being a bachelor. Certainly, an infinite set of prime numbers is not something we can experience via the senses, but we nonetheless know it must exist.
For a long time, the big controversy in philosophy was whether there exists synthetic a priori knowledge. Both examples of a priori knowledge mentioned above are analytic rather than synthetic, i.e. they must be true on pain of contradiction. In contrast, a priori synthetic knowledge is conceivably false but still justified without appeal to sensory experience. This was important because many of our fundamental beliefs about reality are synthetic but appear impossible to justify with sensory experience. For example, a principle of induction seemed indispensable for the justification of scientific theories, yet neither was it true on pain of contradiction nor knowable a posteriori. The problem then, was that if the principle of induction is justifiable, then it must be justifiable a priori, and that would require a priori synthetic knowledge.
In any case, the a priori/a posteriori distinction is not so popular these days. It has been relegated to a pragmatic distinction rather than two fundamentally different kinds of knowledge. The main reason for this has to do with the underdetermination problem. When confronted with a mismatch between our expectations and experience, we must have erred somewhere, but where? The recalcitrant evidence entails that at least some of our assumptions are false, but the arrow of refutation doesn't point to anything in particular. Perhaps, then, so-called a priori knowledge is at fault. For example, there is no purely logical reason why we should not decide to "falsify" the law of excluded middle and reject classical logic in favour of intuitionist logic. In this case, the strong distinction between a priori and a posteriori knowledge is dissolved.
My own take is that the a priori/a posteriori distinction is mostly, but not entirely, pragmatic. However, my reasons are quite different from the more popular argument mentioned above. In fact, I am an non-justificationist. That is, I do not believe that true beliefs are justifiable at all, in any way, shape, or form. Strictly speaking, I deny the existence of both a priori and a posteriori knowledge. Indeed, the most interesting things about knowledge, in my view, are not our subjective beliefs at all, but the objective occurrence of knowledge as an emergent and evolving process.
.
