Short answers, pulled from the story.
Gabriel Marcel, the French Catholic philosopher, coined the term "existentialism" (L'existentialisme) in the mid-1940s. He first applied it to Jean-Paul Sartre at a colloquium in 1945, though Sartre initially rejected the label before publicly adopting it on the 29th of October 1945 in a lecture at the Club Maintenant in Paris.
The phrase, associated primarily with Sartre, holds that human beings arrive in the world without a fixed nature and define themselves through their actions and choices. This contrasts with Aristotle and Aquinas, who taught that a thing's essence is given before its individual existence. For Sartre, there is no inbuilt human purpose; people are free to choose their own.
Existential angst is the negative feeling that arises from the experience of radical human freedom and responsibility. The classic example is standing on a cliff and dreading not only falling but the fact that nothing prevents one from jumping. Unlike fear, which has a specific object, angst has no object, because it stems from the awareness that nothing predetermines one's choices.
Existentialism and nihilism are distinct philosophies, though both are rooted in the experience of a world that appears meaningless. Nihilism dismisses morality and meaning; existentialism, by contrast, insists on persisting through the encounter with absurdity and on creating meaning through choice and action. Albert Camus expressed this in The Myth of Sisyphus (1942) with the line "One must imagine Sisyphus happy."
Bad faith, for Sartre, is the denial of one's own freedom and responsibility. His illustration in Being and Nothingness is a waiter who so thoroughly performs the role of a typical waiter that he avoids acknowledging his freedom to act otherwise. Treating one's choices as inevitable, or pretending that determinism removes personal responsibility, are also forms of bad faith.
Existentialist ideas shaped several strands of psychology and therapy, starting with the work of Otto Rank, Freud's associate for around two decades. Viktor Frankl developed logotherapy, a form of existentialist therapy, and Rollo May's 1958 book Existence introduced European existentialist analysts to American therapists. Existentialist therapists encourage patients to use anxiety constructively rather than suppress it, treating it as evidence of genuine freedom and responsibility.