Given that you have already perused Wiki, have a look at the Stanford Encyclopedia of Philosophy entry here: https://plato.stanford.edu/entries/existentialism/. Colloquially, I shall simply quote myself here: Does the word "absurd" have specialized meaning in philosophical discourse?:
"…existentialists argue that human beings cannot escape asking the question, “What is the meaning of existence?” They deny, however, that there is an answer to this question, and reject every scientific, teleological, metaphysical, or human-created end that purport to provide an adequate answer.
For instance, Sartre posits/presupposes that “existence precedes essence.” Thus while the meaning of life question seeks an a priori metaphysically universal, “why are we here”, “meaning of it all,” answer “from the beyond,” so to speak, no such answer can exist because there is nothing [knowably] there that can provide it (given the presupposition). Or see Camus’ take (from https://plato.stanford.edu/entries/camus/):
Thus, while accepting that human beings inevitably seek to understand
life’s purpose, Camus takes the skeptical position that the natural
world, the universe, and the human enterprise remain silent about any
such purpose. Since existence itself has no meaning, we must learn to
bear an irresolvable emptiness. This paradoxical situation, then,
between our impulse to ask ultimate questions and the impossibility of
achieving any adequate answer, is what Camus calls the absurd. Camus’s
philosophy of the absurd explores the consequences arising from this
Camus’s understanding of absurdity is best captured in an image, not
an argument: of Sisyphus straining to push his rock up the mountain,
watching it roll down, then descending after the rock to begin all
over, in an endless cycle. Like Sisyphus, humans cannot help but
continue to ask after the meaning of life, only to see our answers
tumble back down."
The question "what is an existentialist" is as broad as asking what is an analytic philosopher. The term existentialism (like the term "analytic philosophy") is better conceived as a group of methods, rather than a school of thought, whose focus is the domain of issues that have historically arisen when one finds oneself thrust-into/in existence, and wonders WTF, "why am I here(?)," in the broadest sense.
The question takes as its starting point the experience of the human subject who so finds him/herself— not only the thinking/rational subject, but the acting, feeling, living human individual (i.e. I'm "here," so what now? why so? and how should/do I feel about it,and why so, etc). For the answers of system building existentialist philosophers see Sartre or Heidegger; For Christian existentialists see (especially) Kierkegaard or Jaspers; for a nihilistic existentialist see Nietzsche, for a literary existentialist see Camus, for a sociological existentialist see Ortega-Gasset, etc.