I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using the machinery of etale cohomology. I know a little bit about how etale cohomology groups appear in algebraic number theory but I'd like to know about ways that these things show up in other mathematical subjects as well. Is there anything that an algebraic topologist should really know about etale cohomology? What about a differential geometer?
$\DeclareMathOperator{\gal}{Gal}$ Here's a comment which one can make to differential geometers which at least explains what etale cohomology "does". Given an algebraic variety over the reals, say a smooth one, its complex points are a complex manifold but with a little extra structure: the complex points admit an automorphism coming from complex conjugation. Hence the singular cohomology groups inherit an induced automorphism, which is extra information that is sometimes worth carrying around. In short: the cohomology of an algebraic variety defined over the reals inherits an action of $\gal(\mathbb{C}/\mathbb{R})$. The great thing about etale cohomology is that a number theorist can now do the same trick with algebraic varieties defined over $\mathbb{Q}$. The etale cohomology groups of this variety will have the same dimension as the singular cohomology groups (and are indeed isomorphic to them via a comparison theorem, once the coefficient ring is big enough) but the advantage is that that they inherit a structure of the amazingly rich and complicated group $\gal(\bar{\mathbb{Q}}/\mathbb{Q})$. I've often found that this comment sees off differential geometers, with the thought "well at least I sortof know the point of it now". A differential geometer probably doesn't want to study $\gal(\bar{\mathbb{Q}}/\mathbb{Q})$ though. If I put my Langlandsphilosophy hat on though, I can see a huge motivation for etale cohomology: Langlands says that automorphic forms should give rise to representations of Galois groups, and etale cohomology is a very powerful machine for constructing representations of Galois groups, so that's why I might be interested in it even if I'm not an algebraic geometer. Finally, I guess a much simpler motivating good reason for etale cohomology is that geometry is definitely facilitated when you have cohomology theories around. That much is clear. But if you're doing algebraic geometry over a field that isn't $\mathbb C$ or $\mathbb R$ then classical cohomology theories aren't going to cut it, and the Zariski topology is so awful that you can't use it alone to do geometryyou're going to need some help. Hence etale cohomology, which gives the right answers: e.g. a smooth projective curve over any field has a genus, and etale cohomology is a theory which assigns to it an $H^1$ of dimension $2g$ (<pedant> at least if you use $\ell$adic cohomology for $\ell$ not zero in the field <\pedant>). 


a) Conceptually an algebraic topologist should be interested in étale cohomology, because it answers a very naïve question: given an algebraic variety over $\mathbb C$, how do I calculate algebraically its singular cohomology? The obvious answer "why, I'll just take the cohomology of the constant sheaf $\mathbb Z$ " fails in a spectacular way: if the variety is irreducible ( a reasonable assumption) the cohomology will be zero in positive degree because constant sheaves are flabby, hence acyclic. This is because the Zariski topology of an algebraic variety is too coarse and doesn't allow for the innumerable singular simplices at the algebraic topologist's disposal.Of course you might say "who cares? I'll just do it my singular way " but then what is to be done in nonzero characteristic? This is where Grothendieck's étale cohomology comes in : it allows one to compute a most reasonable cohomology for constant sheaves. And for sheaves of finite abelian groups over complex varieties, a difficult theorem ( aptly named comparison theorem) proved by Mike Artin says that étale cohomology coincides with singular cohomology. b)You ask : "...what other problems one can solve using the machinery of etale cohomology?" Well, there is the proof ( which won him a Fields medal) by Voevodski of Milnor's conjecture on quadratic forms, which had been the preeminent open problem in quadratic form theory for 30 years. He introduced $\mathbb A^1$homotopy which opened a quite active field of research by your algebraic topology colleagues. Here is a survey article by Fabien Morel on $\mathbb A^1$ algebraic topology: http://www.mathematik.unimuenchen.de/~morel/ICMfinal1.pdf Finally, talking of references, an excellent and very userfriendly set of notes by Milne on étale cohomology can be freely downloaded from 


This review by Bloch answers the first part of your question. Algebraic topology like Sullivan's "Geometric Topology" make use of etale cohomology and etale homotopy. 


One of the annoying aspects with sheaf cohomology in algebraic geometry is that  even for curves over the complex numbers  the cohomological dimensions are not what you want them to be, if you are use to the Betti cohomology. Etale cohomology fixes this problem by defining a cohomology from the covers of a space. This is a standard algebrotopological phenomena: e.g. the fundamental group of a space is indeed the direct limit on the deck groups of covers of the space. I don't know if this helps any computations in algebraic topology though. 

