Russell's Paradox is a good example. Russell poses the question: "If R is a set of all sets which don't contain themselves, does R contain itself?" If it does, then it doesn't. If it doesn't, then it does.
Something too damn complicated to explain and therefore irrelevant, especially when someone uses it all the time without actually knowing what it means.
A catch-22 of sorts. Basically, it means that something counterintuitive to common sense's dictations could be potentially true as well....or in layman's terms, a no-win situation.