Catch-22 is a famous anti-war novel written by Joseph Heller. In the book, a catch-22 refers to the law that allows soldiers to raid an Italian whorehouse. The very same law that gives them the right to make this intrusion also entitles them to conceal why they are raiding it in a first place, so the proprieter can never find out why it happened. This is compared to an earlier reference in the book, where a military policy requires that soldiers can only be discharged for insanity if they are both insane and ask for discharge. However, asking for discharge if you are insane is a sane act, so you cannot be insane. Therefore, getting a discharge for insanity is actually impossible.
A catch-22 is commonly misperceived to be a situation wherein both options are seen to have negative consequences. That situation is actually called a double-edged sword. A catch-22, as in the examples above, is a situation wherein the solution to a problem is impossible given the very nature of the problem.
My university has asked me to transfer because they expel anyone with a GPA under a 2.0, yet I cannot transfer because no other college will take an applicant with less than a 2.0! This policy is such a catch-22!