Distance fog is a technique used to disguise the limited rendering distance that was required to make early 3D games run at acceptable frame rates.
The shift to 3D was a very important moment in gaming history, with consoles like the PlayStation and the Nintendo 64 making 3D gaming mainstream. However, 3D games quickly encountered a big problem that wasn't present in 2D games.
Due to hardware limitations, early 3D games can only render about a few hundred polygons in the world, meaning only a limited amount of the three-dimensional world can be rendered at once to keep the frame rate at an acceptable level. Some objects would suddenly appear or disappear if they got close enough or were too far away from the camera to reduce the amount of polygons to render. Likewise, if the player looks past the "boundary" of the available render distance they'd see an empty void of nothingness.
The "solution" developers came up with for this problem was to put a thick layer of opaque fog (with a translucent front layer) right in front of the draw distance limit, that way objects would pop in inside the fog without players noticing it and would emerge from the fog fully rendered. This tactic was more commonly used on Nintendo 64 games due to the system's extremely limited texture cache size (just 4 kilobytes, or about the size of an emoticon).
While this "distance fog" was effective in hiding the draw distance problem, it produces a new problem of the fog blurring out objects and make navigation more difficult because you can only see so much ahead. Superman 64 is notorious for having very noticeable distance fog, as are the first three games in the Turok series: the first in particular is extremely frustrating due to having hitscan enemies which can attack from outside the draw distance. However, some games like Silent Hill were able to use distance fog to their benefit, in this instance to create a more creepy atmosphere.
The validity of distance fog has become a topic of debate in retrospective. Some games like Crash Bandicoot used clever camera angles so that the visible world would never have forced too many polygons to be rendered and never reached the draw distance limit, while other games like Banjo-Kazooie outright omitted using Distance Fog altogether while still being able to render the whole 3D world without problems. The Spyro the Dragon trilogy, Super Mario 64 and Crash Bandicoot: Warped also used a technique called "dynamic LOD" ( "Level of Detail"), which rendered faraway objects with less detail and polygons to improve performance and allow nearby objects to be rendered with more detail, a technique that many games have also used, even to this day.
As newer gaming generations arrived, hardware continues to become more and more powerful, thus allowing for more impressive draw distances with tens of thousands of polygons rendered at once with no problem, thus making distance fog obsolete.