Samuel Beckett's "Waiting for Godot" is the very essence of existentialism (a categorization the author himself would have rejected) and its history is born of Beckett's intense interest in psychoanalysis, both as patient and student. Having said that, little is actually known about the history of the play beyond the mere fact that Beckett wrote it in the early 1950s with its stage debut in Paris in 1953. It is possible that the content of "Waiting for Godot" was in some way influenced by Beckett's World War II experiences serving as a courier with the French Resistance, which entailed regularly evading arrest at the hands of the German Gestapo. Such a connection, however, is purely speculative.
With regard to early audience reactions, they ranged from bored and befuddled to enchanted and intellectually-challenged. "Waiting for Godot" was unlike anything most theatergoers had ever experienced: two tramps or hobos discussing the nature of existence while waiting endlessly for someone who never shows up and probably never will. The entrance of Pozzo and Lucky was seen, unsurprisngly, as dehumanizing, and much of the dialogue between Vladimir and Estragon is depressing. A 1996 biography of Beckett by James Knowlson cited a newspaper review from 1953 describing the reaction of many in the audience as derisive and angry. Knowlson's biography also quotes one of the actors involved in a London production of the play:
"Waves of hostility came whirling over the footlights, and the mass exodus, which was to form such a feature of the run of the piece, started quite soon after the curtain had risen."
Certainly, the structure and content are entirely unconventional, which was discomforting to some early viewers. Even Beckett claimed not to know much about his own characters and why (and, in the case of Godot, whether) they even exist. The play continues to be produced around the world more than fifty years after its' premier, and audiences continue to purchase tickets to see it, so clearly many theatergoers enjoy the challenge of trying to fully understand it.
As much as audience reactions have run the spectrum of possible visceral reactions, so have the interpretations of the play. That is a subject beyond the scope of this answer, but it goes to the heart of the matter. Many people did not and still do not entirely know what to make of "Waiting for Godot." Unlike presentations during the 1950s, however, over time more and more people going to see it have enjoyed the benefit of knowing what they're about to experience.