Some books about existentialism never try to answer this question, as it is hard to define. The key thinkers disagreed so much that, whatever you say, you are bound to misrepresent or exclude someone. Moreover, it is unclear who was an existentialist and who was not. Sartre and Beauvoir were among the very few to accept the label, and even they were reluctant at first. Others refused it, often rightly. Some of the main thinkers in this book were phenomenologists but not existentialists at all (Husserl, Merleau-Ponty), or existentialists but not phenomenologists (Kierkegaard); some were neither (Camus), and some used to be one or both but then changed their minds (Levinas).
All the same, here is my attempt at a definition of what existentialists do. I put it here for reference, but by all means skip it and come back if the need or want arises.
— Existentialists concern themselves with individual, concrete human existence.
— They consider human existence different from the kind of being other things have. Other entities are what they are, but as a human I am whatever I choose to make of myself at every moment. I am free —
— and therefore I’m responsible for everything I do, a dizzying fact which causes
— an anxiety inseparable from human existence itself.
— On the other hand, I am only free within situations, which can include factors in my own biology and psychology as well as physical, historical and social variables of the world into which I have been thrown.
— Despite the limitations, I always want more: I am passionately involved in personal projects of all kinds.
— Human existence is thus ambiguous: at once boxed in by borders and yet transcendent and exhilarating.
— An existentialist who is also phenomenological provides no easy rules for dealing with this condition, but instead concentrates on describing lived experience as it presents itself.
— By describing experience well, he or she hopes to understand this existence and awaken us to ways of living more authentic lives.
Sarah Bakewell, “At the existentialist café”, pp. 33-34.