“If in the first act you have hung a pistol on the wall, then in the following one it should be fired. Otherwise don’t put it there.” – Anton Chekhov
Chekhov’s gun is actually a principle. And it says you should use what you have, otherwise, don’t bother introducing it. In drama this creates a sense of continuity. For example, let’s say there’s a scene where one of the characters mentions that they were an Eagle Scout. It’s useful character information, as it tells us about the person. But you can also be sure, if the show is well-written, that the knowledge this person gained in Eagle Scouts will become useful.
Chekhov’s gun will quite often literally be a gun, though it could be anything: a pill bottle, a jigsaw puzzle, a fireplace poker… etc.
It’s a matter or dramatic satisfaction, and also a matter of economy: introducing things that have no place is a waste of time.
However, if those things return later? Then not only do they have a place: because those elements are already familiar to the audience, they create a stronger sense of narrative satisfaction.
It is true in drama (where Chekhov’s Gun is an established principle.) It is also true in improv.
If two players are improvising a scene where they are snowbound in a mountain cabin, and they start composing a letter which they throw out the window. Well, that act will probably figure into their eventual escape. Who knows how, but that letter becomes a loaded Chekhov’s gun. So don’t ignore it.
If this seems simply like an esoteric yet interesting dramatic truism, you should be aware that this principle also works in conversation.