Existentialism is a philosophical and cultural trend associated with nineteenth-century writers like Søren Kierkegaard, Fyodor Dostoevsky and Friedrich Nietzsche, but which became highly influential in the 1940s, 50s and 60s. Many philosophers, writers and theologians (such as Martin Buber, Franz Kafka, Rudolf Bultmann, Paul Tillich, Jean-Paul Sartre and Albert Camus) include existential themes and ideas in their works.
Other 'existentialists' are Karl Jaspers, Gabriel Marcel, Martin Heidegger and Maurice Merleau-Ponty. The definition "is as fluid as the true existentialist life - unbound, self-sufficient, responsible to no one" .
Existentialism lacks a central doctrine or philosophy, but there are certain ideas which run through existentialist thought, including: doubts about the value of objective study and systems of ideology and belief (including political ideologies and religions), conception of human life and existence as being a 'work in progress', not fixed or often even finished (except by death), and a belief in decisions made in "the moment" taking priority over other aspects of human life. The phrase "existence precedes essence" was used by Sartre as something of a description of the basic ideas of existentialism - that human life can not be abstracted into an 'essence', but the very essence of what it is to be human is decided by the individual's own existence.
In Being and Nothingness (1943) Sartre offers a kind of definition of existentialism saying: "The Nature of consciousness simultaneously is to be what is not and not to be what it is" And hence we come back to our own natures, our own 'essences'. We exist, yes, but how do we 'define ourselves'? It is here that his well-known example of the waiter comes in,
His movement is quick and forward, a little too precise, a little too rapid. He comes toward the patrons with a step a little too quick. He bends forward a little too eagerly; his voice, his eyes express an interest a little too solicitous for the order of the customer. Finally there he returns, trying to imitate in his walk the inflexible stiffness of some kind of automaton while carrying his tray with the recklessness of a tight-rope walker by putting it in a perpetually unstable, perpetually broken equilibrium which he perpetually re-establishes by a light movement of the hand and arm. 
Curiously enough, another book that came out in 1943, She Came to Stay, by Sartre's lifelong intellectual confidant and companion, Simone de Beauvoir, also describes various kinds of consciousness, in passages ranging from wandering through an empty theatre (the stage, the walls, the chairs, unable to come alive until there is an audience) to watching a woman in a restaurant ignore the fact that her male companion has begun stroking her arm ("it lay there, forgotten, ignored, the man's hand was stroking a piece of flesh that no longer belonged to anyone")
De Beauvoir, unlike Sartre, would have been aware that many of existentialism's elements, for example, the notion of 'the Other', can be found in Hegel, where they were in turn borrowed from the Eastern tradition, with its 'de-emphasis' of individualism as a delusion born of ignorance - and perhaps conceit. 
The difficulty comes for existentialism in that if you accept that existence precedes essence, how can we judge whether one has a good life? Most philosophies have a built-in idea of the good life - reaching an enlightenment or eudamonia, salvation and union with God, the achievement of the Communist ideal, preparedness for death - but if you reject the philosophical or theological essences underpinning these, what criteria for success do we have? Existentialism proposes that one achieves a freedom through 'authenticity', being the author and maker of one's own life.
Existentialism was influential on art and cinema - with the films of Ingmar Bergman and the paintings of Abstract Expressionists like Jackson Pollock often being understood in the language of existentialism. Much of the critical theory that came to dominate Continental philosophy and culture has its roots in existentialism.
- There is a description of Wols' life here: http://paintings.name/paintings-art-informel.php
- 'Essentials of Philosophy and Ethics', edited by Martin Cohen, Hodder Arnold 2006 p.94
- Being and Nothingness, by Jean Paul Sartre (1943) see for example http://cbae.nmsu.edu/~dboje/teaching/503/sartre_links.htm
- Philosophical Tales, by Martin Cohen, Blackwell 2008 p 244