Post-Structuralism is a body of work that is a response to structuralism; it rejects structuralism yet for various reasons still defines itself in relation to it. So the best way to understand post-structuralism is to understand structuralism.
Structuralism, considered broadly, is any theory that follows Immanuel Kant's notion that the mind actively structures perceptions (Jean Piaget and Noam Chomsky are structuralists in this sense), or any theory that follows Durkheim's attention to social structure (e.g. classifying societies as mechanical or organic). More narrowly, structuralism is inspired by the work of the linguists Roman Jakobson and Ferdinand de Saussure. Their main argument is that language is not just a set of words (abstract) that refer to things (concrete). In other words, the word "rock" does not have sense, or meaning, simply because we identify it with real rocks. Rather, language consists of a system of meaning; that is, the meaning of any one word is determined by its relationship with other words (thus, a dictionary doesn't juxtapose words with pictures of things; rather, it defines words in terms of other words). When looking solely at language or systems of meaning that function linguistically, this approach is called semiotics. When looking at other phenomena, it is structuralism. In short, any approach that sees the meaning of something as subordinate to its place within a system is structuralism. The most important structuralists were French scholars who tried to adapt these principles to other fields of study: the psychoanalyst Lacan, the philosopher Louis Althusser, and the anthropologist Claude Levi-Strauss.
Note: structuralism is in many ways opposed to humanism, because it privileges structures and systems over the specific parts of these systems (e.g. actual humans).
Note: structuralism requires some space between the system and the person studying the system -- in other words, structuralism is a way of studying structures objectively.
Post-structuralists are quite simply all people who take structuralism very seriously and recognize its importance, yet on some level profoundly disagree with or even actively reject it. This ambivalence echoes a deeper ambivalence towards the whole Enlightenment project. Like Kant and his contemporaries and successors, they believe in the importance of critical thinking (the philosopher Jürgen Habermas is probably the most important heir to Kant today -- not that he is strictly speaking a "Kantian", but in a more general sense that he believes that through reason we can understand the world and achieve enlightenment). Unlike Kant and his successors, they are highly skeptical of progress. You might say they take Kant's critical approach one step further by turning it against itself, and thus criticizing the Enlightenment assumptions that objectivity is possible and good, and that the positive accumulation of objective knowledge is possible and good. They are so true to this critical spirit that, unlike post-modernists, they do not whole-heartedly celebrate the demise of the Enlightenment project. (In this ambivalent turn they are something like contemporary heirs to Nietzsche, and many explicitly refer to Nietzsche for inspiration, even if they do not agree with everything he wrote.)
Other than a disagreement with the tenets of structualism, many post-structuralists are sharply critical of one another. This is one reason why a group with such divergent views are called post-structuralists and not something else - once you get beyond their debt to structuralism and the fact that they nevertheless are not structuralists, there is nothing else to define them as a group. The most famous post-structuralists - although they express often fundamentally divergent views - are the philosopher Jacques Derrida, the historian Michel Foucault, and the sociologists Pierre Bourdieu and Bruno Latour.
See also