How does "Waiting for Godot" relate to ways of thinking and how does it represent existentialism?
Tagging along with akannan's post: Existentialism means that existence precedes essence. So, your essence (who you are, what you think, and how you choose to interact with others and the world) is created manifested after you exist. You are not predetermined (aside from genetic predispositions, etc.). Your essence is your creation. There is great freedom there, but also great responsibility and with that responsibility, there can be anxiety because the choices you make manifest that essence, or meaning in life.
In Waiting for Godot, Vladimir and Estragon (and the audience) are duped, thinking that the climax, epiphany, or any significant meaning will come when Godot arrives. To the contrary, the meaning is the interaction of the characters as they wait. They are waiting for someone or some sign to give some message; when, existentially speaking, it is the waiting - from birth to death - that is life itself. The existentialist would treat the waiting as the significance and meaning in life: the degree and profundity of that meaning is up to each individual. Vladimir and Estragon are waiting, even clinging to an idea of something more, a guide; instead, they should be Godot: in other words, they should take responsibility for their own happiness.
Beckett's work can be seen as existentialist because it provides little in the way of "bad faith" resolutions or totality. There is a lack of any notion of coherent visions which seek to provide a reductive answer to consciousness and being in the world. The characters do not experience massive epiphanies or transformations which allow a totalizing arc to happen, nor is there any narrator or third person omniscient vision to give structure. Rather, they talk and interact, speak and hear, torment and accept abuse entirely on their own. Even at the end, when told that Godot is not coming, Vladimir and Estragon remain paralyzed in their wait. The play is existentialist because, like us, it exists with little semblance of structure where existence precedes essence.