Dilemma
Etymology
From Greek "di-" (two) and "lēmma" (premise, thing taken), from "lambanein" (to take). It began as a term of formal logic — an argument forcing you onto one of two horns — which is why we still speak of being caught "on the horns of a dilemma."
noun · /dih-LEM-uh/
A choice between two options, each of which is unwelcome or difficult.
What it names
It marks the specific trap of two bad roads and no third — distinct from an ordinary problem, which usually has a discoverable good answer.
Example
The dilemma was real: report the friend and lose the friendship, or stay silent and share the guilt.
Related idea
A dilemma has exactly two poor options; a predicament is any troublesome situation, and a paradox is a statement that defies resolution rather than a choice that demands one.