Existentialism is a huge subject in philosophy. Um. In very broad strokes, it deals with the human subject and the human condition, life and death, the search for meaning, the role of the emotional and irrational side of our nature. I think the Wikipedia article gives some very good touchstone information to spring off on. Kierkegaard’s angst, Sarte’s radical freedom, Camus’ absurdity, Nietzche’s will to power.
Sarte’s slogan “existence precedes essence” is often used to describe the core of existential thought. Again, the Wikipedia article gives a good surface glance at what Sarte meant. It’s a rejection of a lot of the previous philosopies, and an assertion that human beings are individuals and have the power to make their own meaning, in various ways. (That’s always what I took it as, searches on “existence precedes essence” will probably yield a ton of ink spilled on the subject) The classic image of two people staring up at the stars and asking “why are we here?” is classic existentialism.
The history of thought that came before existentialism, and how the forerunners in the 19th century laid the foundations of it is really very fascinating. It’s a rejection of rationalism, of positivism and scientism, basically the Enlightenment thinking. It was influenced by the horrible destruction of both world wars, accompanied by a sort of pessimistic outlook… I could go on and on. I’m such a nerd. -_-
I am not a theologian but I’m pretty sure Catholic thinking is not friendly to a great deal of existential thinking. (There are probably points that line up, though) If I recall, Kierkegaard was a Christian, but Sarte wasn’t. And Nietzche had nothing good to say about Christianity. I’m sure there’s ways to reconsile existential thought with Catholic teaching. I’m just not familiar with it.