A dilemma is “a situation necessitating a choice between two equal, esp. equally undesirable, alternatives”, or “a problem that seems incapable of a solution” [source].
It comes, via Late Latin, from the Ancient Greek δίλημμα (dílēmma, – ambiguous proposition), from δι- (di-, having two of) and λῆμμα (lêmma, – premise, proposition) [source].
Today I spotted the word trilemma in an article in The Spectator. I hadn’t seen it before, but from the context it appears to be a variant of dilemma involving three choices.
According to Wiktionary, a trilemma is “A circumstance in which a choice must be made between three options that seem equally undesirable” or “put another way, in which a choice must be made among three desirable options, only two of which are possible at the same time.”
I thought trilemma was a recently-coined word, but according to Wikpedia, it was first used in writing back in 1672.