Existentialism is often described by its questions rather than its method, but the method is what let philosophers as different as Kierkegaard, Sartre and Heidegger be read as one movement at all. Edmund Husserl founded phenomenology at the turn of the twentieth century as a discipline for describing experience exactly as it presents itself to consciousness, bracketing off the question of what really exists behind the appearance in order to examine the appearing itself with rigor. Martin Heidegger, Husserl's student, redirected that method away from consciousness in the abstract and toward Dasein, his term for the particular kind of being a human is, one that always finds itself already thrown into a world, concerned about its own existence, and moving toward its own death.
Being and Time, published in 1927, gave existentialism most of its working vocabulary before the word itself was in wide use: thrownness, being toward death, anxiety as the mood that reveals existence stripped of its everyday cover, and authenticity as the demand to take ownership of a life one did not choose to begin. Jean-Paul Sartre read Heidegger in a German prisoner of war camp and carried this apparatus into Being and Nothingness and into a public philosophy accessible well outside the seminar room. What phenomenology supplied was not a set of conclusions, since Husserl, Heidegger and Sartre disagreed sharply about most of them, but a shared insistence on starting from lived, first person experience rather than from abstract theories about human nature, which is the thread that lets a religious Dane and an atheist Frenchman count as the same movement.