A priori and a posteriori refer to two different types of knowledge, or alternatively to two different ways in which beliefs can be justified. A priori knowledge acquires its justification prior to, or independently of, any observational or empirical evidence (e.g. scientific evidence, simply looking and observing something in the world, etc). A posteriori knowledge, by contrast, is justified only after, or as a result of, observational or empirical evidence. Mathematical truths, tautologies (such as ‘a bachelor is unmarried’), and logical deductions from other a priori knowledge are commonly cited examples of beliefs that can be justified a priori, without empirical evidence. Most other types of belief, including personal experiences, historical knowledge, scientific knowledge, etc, are justified a posteriori, through appeal to empirical evidence of some sort.
The basic distinction between a priori and a posteriori knowledge essentially rests in whether or not a given thing can be known by ‘pure thought’, in which case it is knowable a priori. Conversely, if the input of the senses is needed, then the belief in question is only knowable a posteriori. Some philosophers have questioned the validity of the a priori/a posteriori distinction, arguing that even mathematics and logic have an empirical or experiential component. Notwithstanding these controversies, the conceptual distinction is an important one to be familiar with when engaging in philosophical discourse and evaluating evidence.
A priori vs a posteriori reasoning: useful introductory video
A priori and a posteriori: a useful discussion of the basis of the distinction from New World Encyclopedia
A priori: definition and examples of a priori knowledge from philosophy-index.com
A posteriori: definition and examples of a posteriori knowledge from philosophy-index.com
A priori justification and knowledge: detailed analysis of the concept of a priori knowledge from the Stanford Encyclopedia of Philosophy
A priori and a posteriori: Internet Encyclopedia of Philosophy overview article