Chekhov's Gun is an object that a writer describes early on in a work of fiction or script that later becomes a focus point for action. The object could be an actual gun sitting on a table or a sword hanging on a wall.
Anton Chekhov thought that if an author spent time describing such an object it should show up later in the story. At the same time Chekhov also thought it was wrong to spend time describing or showing an object without later referencing it in the story or play.
Chekhov once wrote in a letter, "One must never place a loaded rifle on the stage if it isn't going to go off. It's wrong to make promises you don't mean to keep." There are variations on precisely how Chekhov stated this as the Wikipedia entry explains. The concept remains that same in each variation.
Chekhov's Guns appear frequently on television and in movies. They are so commonplace that some television viewers look for them. This has led to the use of false Chekhov's guns to mislead viewers. Sometimes a Chekhov gun object is not revisited due to a mistake. The writers may have wanted to set up an item but later decided not to use it. It could also be that a series changed writers and a Chekhov's gun was forgotten about.
TV Tropes has a list of objects and things that when referenced on a TV show or film often foretell of some later action, such as when a character is coughing or an empty pool is shown.
This video essay from HoustonProductions1 provides several examples of Chekhov's Guns through setups and payoffs in movies.