WHO coined the term existentialism and what is it?

Jean-Paul Sartre coined the term, though the concepts were in discussion for nearly a century prior by men like Dostoevsky, Nietzsche, Heidegger, Camus, and Kafka, among many others. It is essentially the belief that life has no higher purpose, no inherent value, and that it is the duty of every man to give his own life purpose and define his own values.

I highly recommend Sartre's "Existentialism is a Humanism" - it's a short, concise snapshot of existentialism as a belief system.