Why étale?
Background: The notion of an étale morphism has proved itself to be ubiquitous within the realm of algebraic geometry. Apart from carrying a rich intuitive idea, it is the first ingredient in notions and theories such as étale cohomology, a Galois theory of schemes, and algebraic spaces. It is safe to say that, of the many classes of morphisms that we care for in algebraic geometry, étale morphisms are among the most important ones.
Question: How is it a priori clear that étale morphisms are important, and in particular, how can we see that étale morphisms are often 'the right' thing to look at?
Example: One could define variants of étale cohomology by redefining the étale site using slightly different classes of morphisms. In some ways this has been done, as things like flat cohomology and Nisnevich cohomology are a thing; however, étale cohomology is arguably more important than these variants. What is it that distinguishes étale cohomology here?
Example: We could reconsider Galois theory by replacing 'étale' with something else, and see what happens. Probably this does not lead anywhere (as otherwise someone would've written about it). Perhaps, we could argue here that in the differential-geometric case, one loses all hopes of a classification if one replaces 'covering space' by something slightly weaker. But I'm not entirely convinced by this.
What I know: I know that people love to tell the story of how the Zariski topology does not have enough opens and so we want more. However, this does not tell us why we specifically choose étale morphisms rather than something similar. I also know that, to emphasize the strength of the étale site, people say that the étale site gives us an algebro-geometric analogue on an Implicit Function Theory, however I've never seen this explicitly in action, nor do I think it can explain everything.
Solution 1:
Let $f:X \to Y$ be a morphism---say, for simplicity---of finite type between Noetherian schemes. Thus, we may think of $f$---i.e. of the $Y$-scheme $X$---as defining a contravariant functor $\Phi$ on the category of Noetherian $Y$-schemes.
Then to say that $f$ is étale is equivalent to saying that this functor $\Phi$ is immune/blind to nilpotent thickenings, i.e. that if $S_0$ is a closed subscheme of a Noetherian $Y$-scheme $S$ defined by a nilpotent sheaf of ideals on $S$, then the map$$\Phi(S) \to \Phi(S_0)$$---induced by the natural inclusion of $S_0$ into $S$---is a bijection.
This condition of being blind to nilpotent thickenings is natural in the sense that it is natural to expect that any sort of "natural topology" on $Y$ should be blind to nilpotent thickenings. Put another way, we may think of an étale morphism as a morphism that is "orthogonal to nilpotent thickenings".
One closely related property of étale morphisms is the invariance of the étale site with respect to nilpotent thickenings, as well as with respect to morphisms such as the Frobenius morphism in positive characteristic. It is natural to expect such invariance properties to hold since it is natural to expect that any sort of "natural topology" should be blind to nilpotent thickenings or to the Frobenius morphism in positive characteristic.