Post-apocalyptic video games are part of a subgenre of science fiction, fantasy, and horror media that spans multiple categories of game from role-playing games and first-person shooters to survival horror and simulations.
The best way to think about post-apocalyptic video games is as a descriptor of the setting of the video game and less than the particular subgenre of a title given that a post-apocalyptic video game can literally be any type of game, including even puzzle games.
These games tend to feature a narrative that focuses on the aftermath of a cataclysmic event which can be anything from a population-decimating plague to nuclear war and the ever-popular zombie outbreak or the always reliable asteroid colliding into Earth. Typically the environments are bleak and depressing with civilization scattered to nonexistent.
A particularly strong theme in the Fallout series of role-playing games, post-apocalyptic video games often pit different factions of survivors against one another, each motivated by their own aims, some of which are rational while others are inscrutable to our sensibilities. Post-apocalyptic video games most of the time warp or distort the culture of the pre-disaster civilization, often leading to familiar yet vastly different organizations. Classic examples of this include the doomsday cult that thinks they called it right to the military holdouts on a base in the hinterlands.
A universal theme found in all post-apocalyptic video games is that civilization is a precious, fragile construct that is only held together by delicate threads that are easily sundered when necessity calls for it.