What is “Chekhov’s Gun” and Why Does it Matter?

Have you heard the term "Chekhov's gun" before, when talking about a scene from a story? Maybe it was in the context of a negative review of a movie or show you just watched, or maybe someone critiquing your work suggested there's a case of "Chekhov's gun" that you should address. So what is it, exactly, and how can you spot it from now on?