A priori knowledge is knowledge gained or justified by reason alone, without the direct or indirect influence of experience (here, experience usually means observation of the world through sense perception.)
A posteriori knowledge is any other sort of knowledge, viz. knowledge the attainment or justification of which requires reference to experience. This is also called empirical knowledge.
One of the fundamental questions in epistemology is whether there is any non-trivial a priori knowledge. Generally speaking rationalists believe that there is, while empiricists believe that all knowledge is ultimately derived from some kind of external experience.
The fields of knowledge most often suggested as having a priori status are logic and mathematics, which deal primarily with abstract, formal objects.
Empiricists have traditionally denied that even these fields could be a priori knowledge. Two common arguments are that these sorts of knowledge can only be derived from experience (as John Stuart Mill argued), and that they do not constitute "real" knowledge (as David Hume argued).
Note that discussions of a priori and a posteriori knowledge almost always concern propositional knowledge, or, roughly, "knowledge that". The authors don't know if the distinction has found its way into discussions of other kinds of knowledge, e.g. "knowledge how". (See Knowledge for more information on the that/how distinction.)