Lyotard supplied the term the period is known by, in a book he later said was the worst thing he wrote.
The Postmodern Condition (1979) was commissioned as a report on knowledge for the government of Quebec, and its argument is compact: the sciences have always legitimated themselves by appeal to a metanarrative — the emancipation of humanity through reason, the realization of spirit, the liberation of the worker — and those stories have lost credibility. What replaces them is not a better story but a plurality of local language games, each with its own rules of validity and no master discourse able to adjudicate between them. Knowledge becomes something produced and exchanged rather than something contemplated, and its criterion shifts toward performativity: what counts is what works.
He drew a political conclusion the label is rarely credited with. Lyotard had spent the 1950s and 60s in Socialisme ou Barbarie and in the Algerian struggle, and his suspicion of grand narratives came from watching one — Marxism — justify what was done in its name. The Differend (1983) is the more serious book: it concerns cases where a wrong cannot be expressed in the idiom of the tribunal that would judge it, so that to be heard at all the injured party must accept terms that have already conceded the case. Injustice on this account is not merely unequal treatment but the absence of a language in which the harm can be phrased.
That is a far more precise position than "postmodernism" as it circulates, and it is why Lyotard is better read as a theorist of incommensurable claims than as a relativist.