The origin of this word is from a novel of the same name, written by Joseph Heller in the 1960s. And it is, by the way, an excellent book! I highly recommend it to anyone with a good English level, though it may be too difficult for learners under a B2.
In a catch-22 the solution to a problem is contingent on another part of the problem. To take an example from the book:
If a fighter pilot wants to go home, he must fly a large number of hours. He wants to go home because he doesn’t want to die in the war. However, the more he flies, the higher his chances are of dying in the war. So he’s in a catch-22. To leave the war he must fly more, but flying more increases his chances of dying in battle, so he should fly less, but that means he can’t go home as soon.
It’s a book about war written as a comedy; I couldn’t put it down!