A "Catch-22" is "a problem for which the only solution is denied by a circumstance inherent in the problem or by a rule." For example, losing something is typically a conventional problem; to solve it, one looks for the lost item until one finds it. But if the thing lost is one's glasses, one cannot see to look for them — a Catch-22. The term "Catch-22" is also used more broadly to mean a tricky problem or a no-win or absurd situation.
In the book, Catch-22 is a military rule typifying bureaucratic operation and reasoning. The rule is not stated in a precise form, but the principal example in the book fits the definition above: If one is crazy, one does not have to fly missions; and one must be crazy to fly. But one has to apply to be excused, and applying demonstrates that one is not crazy. As a result, one must continue flying, either not applying to be excused, or applying and being refused. The narrator explains:
There was only one catch and that was Catch-22, which specified that a concern for one's safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn't, but if he were sane he had to fly them. If he flew them he was crazy and didn't have to, but if he didn't want to he was sane and had to. Yossarian was moved very deeply by the absolute simplicity of this clause of Catch-22 and let out a respectful whistle. (p. 56, ch. 5)
Other forms of Catch-22 are invoked throughout the novel to justify various bureaucratic actions. At one point, victims of harassment by military police quote the MPs' explanation of one of Catch-22's provisions: "Catch-22 states that agents enforcing Catch-22 need not prove that Catch-22 actually contains whatever provision the accused violator is accused of violating." Another character explains: "Catch-22 says they have a right to do anything we can't stop them from doing."
Yossarian comes to realize that Catch-22 does not actually exist, but because the powers that be claim it does, and the world believes it does, it nevertheless has potent effects. Indeed, because it does not exist, there is no way it can be repealed, undone, overthrown, or denounced. The combination of force with specious and spurious legalistic justification is one of the book's primary motifs.
The motif of bureaucratic absurdity is further explored in 1994's Closing Time, Heller's sequel to Catch-22. This darker, slower-paced, apocalyptic novel explores the pre- and post-war lives of some of the major characters in Catch-22, with particular emphasis on the relationship between Yossarian and tail gunner Sammy Singer.