I've been aware of stacks since grad school, and I can usually follow in rough lines a discussion about stacks, but I've often wondered what particular (purely!) scheme-theoretic argument or theorem is significantly simplified by the introduction of stacks. I'm sure there are many, but since I don't deal with stacks on a regular basis I don't encounter them as frequently, and I thought maybe some of you can enlighten me.
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3$\begingroup$ Cohomology of stacks provide useful "algebraic/arithmetic" constructions superior in some respects to what can be done using coarse moduli schemes. For example, if one wants to equip spaces of modular forms with $\mathbf{Z}$-structure in a manner that handles all primes in a unified manner, it can be simpler to work with moduli stacks of (generalized) elliptic curves rather than adjoining "extra level" and using fine moduli schemes. For example "$X_0(N)$" is always a regular proper Artin stack, whereas the coarse moduli scheme cousin is neither regular nor fine (but is also useful!). $\endgroup$– BCnrdCommented Oct 17, 2010 at 1:09
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$\begingroup$ I haven't met stacks often, but I used to think that they are tautological reformulations of: a) moduli problems b) quotient problems, that allow to "speak geometrically" (e.g. a moduli stack has a tangent bundle, a "moduli problem" doesn't) $\endgroup$– QfwfqCommented Oct 17, 2010 at 17:15
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The classic application is Deligne & Mumford's paper proving the irreducibility of the coarse moduli scheme $\overline{M}_g$ of stable genus g curves over any algebraically closed field. They proved irreducibility first for the moduli stack $\overline{\mathcal{M}}_g$ of curves, and then inferred from this result the irreducibility of the scheme.
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The DeRham space is a stack $X_{DR}$ associated to a smooth variety $X$, so that modules on $X_{DR}$ are D-modules on $X$. This is accomplished by declaring the maps from $Y$ into $X_{DR}$ are the same as maps from $Y^{red}$ (the reduced scheme) into $X$. This has the effect of identifying points with their infinitesmal neighborhoods.
The DeRham space is often most useful as a conceptual tool. However, a specific application of it was by Ben-Zvi and Nevins, who used it (and other tools) to show that certain cusped versions $\widetilde{X}$ of $X$ had equivalent categories of D-modules. The idea being, these cusps were identifying some of the infinitesmal neighborhoods of some of the points, and so they should be intermediate between a variety and its DeRham space.