Existentialism is not, of course, a literary movement but a philosophy but is include here as it had a big influence on many writers, particularly certain French writers of the middle of the twentieth century. The basic (very basic) principle of existentialism is, as the name implies, that we should focus on human existence and how we exist in the world. Existence precedes essence, which means that we exist before our nature is determined and it is therefore existence that is primordial. Features of human existence, such as free will, personal responsibility, discipline and choice of options, are key. From the literary point of view, it was this philosophy that informed the writings of certain French writers, such as Jean-Paul Sartre, Simone de Beauvoir and Albert Camus. Their novels dealt with individual responsibility and individual freedom. In some cases – Albert Camus‘ L’Etranger (UK: The Outsider; US: The Stranger) and André Gide‘s Les caves du Vatican (US: The Vatican Swindle; Lafcadio’s Adventures; UK: The Vatican Cellars) are two well-known examples – the idea of an acte gratuit, which badly translates as a gratuitous act (i.e. a spontaneous, gratuitous act, made without any real logic, except doing it for its own sake), was used to push the boundaries of freedom. The whole concept of existentialism was extended by critics to include writers as disparate as Franz Kafka and Dostoyevsky and by journalists to explain student revolt, the events of 1968 and other anti-authoritarian actions.
Note that the links below are only about existentialism and the key French writers associated with it. For more existentialism in general, you could start here or here, though a basic web search will give you any number of hits. For more information on the key French writers, see links above.
Jean-Paul Sartre (1905-1980): Existentialism
Sartre’s lecture Existentialism Is a Humanism
Summary of Some Main Points from Sartre’s Existentialism and Human Emotions
Albert Camus and Existentialism
Camus’s Critiques of Existentialism