Estragon is also called "Gogo," and in one odd moment, "Adam."
At first it seems like Estragon is around to play second fiddle to Vladimir. He comes off as inferior to Vladimir; he has a chronically poor memory, struggles ineptly with his boots, needs to be told what to do (and perhaps even craves such instruction), and spends most of his time trying to fall asleep, unless he’s sleeping already, in which case our guess is he’s trying desperately not to wake up. Or, to make a long story short, he comes across as a simpleton.
Except Estragon isn’t actually a simpleton at all. Sure, he may be lacking in, shall we say, intellectual fortitude (he folds under his own questioning), but he has this habit of tossing out unbelievably profound comments as though they were nothing. But you probably want specifics. To start, look at the "let’s hang ourselves!" exchange in Act I. Estragon realizes right away that the bough might not support Vladimir. Though he asks Vladimir to "use [his] intelligence," he still ends up having to explain the situation to his friend.
More interesting than his foresight, however, is his understanding of the emotional significance of such a consequence: Vladimir would be worse off left alone than he would be dead. In Act II he wowed us with the "We’re all born mad; some remain so," and he displays his own unique reasoning when he asks Pozzo if, having gone blind, he can now see into the future (traits commonly linked in mythology). But Estragon totally takes the cake when he answers one of our biggest questions about Waiting for Godot: why do he and Vladimir fill their time with silly activities and play-acting? "We always find something," Estragon casually remarks in Act II, "to give us the impression we exist."
On top of these moments of profundity, Estragon gets credit for what we think of as the three encapsulating lines of Waiting for Godot. As far as we’re concerned, all the play’s action could be condensed in the following way: 1) "Nothing to be done" (also the opening line of the play), 2) "Nothing is certain," and 3) "Nothing happens, nobody comes, nobody goes, it’s awful!" (By the way, is it coincidence that all three of these start with "Nothing?")
And while we’re on the topic, there’s one more interesting "nothing" line to look at, also courtesy of Estragon: "Don’t let’s do anything. It’s safer." The point is, thematically, this is one of the most important lines in the play. What we were saying is, Estragon’s line justifies his and Vladimir’s inability to choose to act by claiming that doing nothing at all is safer. If you never act, you can never act wrong, and if you never choose, you can never choose incorrectly. The problem is, as a very wise and famous person once said, we choose even when we do not choose; we choose by not choosing. Doing nothing is as unsafe as doing something. And that’s bad news for Estragon.
Did you notice that Vladimir seems to be the connection to Godot, whereas Estragon is along for the ride? Gogo has to be constantly reminded that they’re waiting for Godot, and he has to ask repeatedly who Godot is, what he looks like, etc. It could be that, again, the man is cerebrally challenged. He doesn’t have the intellectual capacity to think about Godot or religion the way that Vladimir does.
But the other possibility is that he has the same profundity when it comes to religion that we’ve seen elsewhere in the play. Just look at his comment on the Bible in response to Vladimir’s inquiry. On the one hand, Estragon looked at the pictures rather than reading the gospels (child-like simplicity). On the other hand, he speaks of "the Holy Land" with a deep emotional understanding of what awaits him in a heaven-like afterlife: "That’s where we’ll go," he says, "we’ll be happy." Notice that Vladimir is bogged down in the logistical inconsistencies from gospel to gospel; more about that later.