A Catch-22 is a trap made of rules. You cannot do the first thing until you do the second, but you cannot do the second until you have done the first. You are stuck, and the stuckness is the design. The American writer Joseph Heller coined the name in 1961 in his novel Catch-22. In the book, a soldier tries to look mad so he will not have to fly more dangerous missions; but wanting to stay alive is taken as proof that he is sane, so they will not let him stop. People now use Catch-22 for any bind like that — jobs, visas, school rules — not only war. You write it with a capital C and a hyphen. It is not a real army rule numbered 22; the number is part of the joke in the novel. Informal writing sometimes drops the capitals, but the idea stays the same.