Postmodernism is a label mostly applied from outside, and the figures gathered under it would have rejected both the term and each other. What they share is a target: the idea that beneath history there is a stable subject, a neutral reason, and a direction of travel.
Lyotard gave the movement its slogan by defining the postmodern as incredulity toward metanarratives, the great legitimating stories of emancipation through reason or through revolution, and arguing that knowledge after them fragments into local language games with no master rule to adjudicate between them. Derrida's deconstruction worked on the texts themselves, showing that the oppositions Western philosophy rests on, speech over writing, presence over absence, nature over culture, are neither stable nor innocent, and that meaning defers endlessly instead of resting on anything final.
Foucault is the politically consequential figure and the reason this tradition matters to anyone outside a literature department. His claim is that knowledge and power are not opposed but produced together. The prison, the clinic, the census and the sexual confession are not neutral instruments applied to pre-existing subjects; they are the machinery that makes those subjects. Power on this account is not held by a sovereign and exercised downward. It is capillary, present in the routines and classifications of ordinary institutions, and it works by producing what counts as normal.
The politics that follows is real and hard to locate on the usual axes. It is corrosive of every settled justification, progressive ones included, which is why the tradition has been attacked from the left as demobilizing and from the right as nihilist. Its actual inheritance is a habit of asking whose interest a description serves, now so widespread that it is used daily by people who would repudiate the source.