I recently tried to explain the fppf site to a differential geometer. I started with the etale site, where I had two motivating claims:
If X is a smooth projective variety over the complexes, the etale topology "recovers" the traditional topology. More precisely, the category of sheaves on the etale site and the category of sheaves on the complex-analytic site are equivalent. (EDIT: I have led my friend astray! See below.)
With just three words scare-quoted: The etale topology is (by "definition") the coarsest "topology" that makes the inverse function theorem "true."
I don't know how to make a statement analogous to (1) or (2) for the fpqc (or fppf, or ...) topology. I don't even really have a feeling for the difference between these topologies. Is there an analogue of (1) or (2) or anything else "purely geometric" to hang my hat on?