Post-Apocalyptic is a form of Setting Aesthetics that, as its name suggests, takes place in a once-thriving world that was subsequently blown up or otherwise ravaged by massive disasters of some kind or another. It's an extremely popular aesthetic with many different forms, ranging from nuclear war to zombie apocalypse to alien invasion, and has been an amendment to or basis for many different /tg/ related publications.
Many Post-Apocalyptic settings has the cataclysm as its main theme - Mad Max and Fallout are good examples of worlds where the apocalypse is in the forefront - but others are more subtle. Warhammer 40,000 is technically post-apocalyptic, as is Age of Sigmar. Dark Souls is also an example, where the apocalypse is there but it isn't clearly about an apocalypse as much as its about just doing things in a weird world.
Most Post-Apocalyptic media is also Sci-Fi, since the cataclysm that started them required higher level technology, but lower level players are often stuck with “current” or even primitive technology. There’s a rather decent rate of crossover with Mecha, with many prominent works of that series being post-apoc since it allows for high enough technology without the many things that should render them useless. Not all Post-Apocalyptic settings are sci-fi ones however, as settings like Dark Sun having a magically induced apocalypse.