On the Wikipedia page Sheaf, under the section "The etale space of a sheaf," the author claims that the etale space of the sheaf of (continuous) sections of a continuous map $Y \to X$ is (homeomorphic to) $Y$ if and only if $Y \to X$ is a covering map. This doesn't seem quite right to me. Isn't the appropriate condition weaker? I think that it should be something like a local homeomorphism with discrete fibers, unless I misunderstood the Wikipedia article. Am I correct?
You are right. The condition is that $Y\to X$ is an etale map, i.e. a local homeomorphism. (it is the standard equivalence between sheaves and etale maps)