Although often treated like a philosophical school of thought, it would be more accurate to describe existentialism as a trend or tendency that can be found throughout the history of philosophy. If existentialism were a theory, it would be unusual in that it would be a theory that is generally opposed to philosophical theories.
More specifically, existentialism displays hostility towards abstract theories or systems that propose to describe all of the intricacies and difficulties of human life through more-or-less simplistic formulas. Such abstract systems tend to obscure the fact that life is actually a rather rough-and-tumble affair, often very messy and problematic. For existentialists, there is no single theory that can contain the whole of the experience of human life.
It is the experience of life, however, which is the point of life - so why isn't it also the point of philosophy? Over the course of millennia Western philosophy has become increasingly abstract and increasingly removed from the lives of real human beings. In dealing with technical issues like the nature of truth or knowledge, human beings have been pushed further into the background. In constructing complex philosophical systems, no room is left for real people anymore.
That is why existentialists focus primarily on matters such as choice, individuality, subjectivity, freedom, and the nature of existence itself. The issues addressed in existentialist philosophy involve the problems of making free choices, of taking responsibility for what we choose, of overcoming alienation from our lives, and so forth.
A self-conscious existentialist movement developed first in early twentieth century Europe. After so many wars and so much devastation throughout European history, intellectual life had become rather drained and tired, so it should not have been unexpected that people would have turned from abstract systems back to individual human lives - the sorts of lives that had been dehumanized in the wars themselves.
Even religion no longer held the luster it once did, failing not only to provide sense and meaning to people's lives but even failing to provide basic structure to daily living. Both the irrational wars and the rationalized sciences combined to undermine people's confidence in traditional religious faith - but few were willing to replace religion with secular beliefs or science.
As a consequence, there developed both religious and atheistic strands of existentialism. The two disagreed on the existence of God and the nature of religion, but they did agree on other matters. For example, they agreed that traditional philosophy and theology had become too remote from normal human life to be of much use. They also rejected the creation of abstract systems as a valid means of understanding authentic modes of living.
Whatever "existence" is supposed to be, it isn't something that a person will come to understand through intellectual posturing; no, the irreducible and undefinable existence is something that we must encounter and engage through actually living. After all, we humans do define who we are through living our lives - our natures are not defined and fixed at the moment of conception or birth. Just what constitutes an "actual" and "authentic" mode of living, though, is what many existentialist philosophers tried to describe and debated about with each other.