A warp zone is usually an area in a video game where players can go from one place or level to another. They are sometimes used as cheats (Super Mario Bros.) and sometimes as ways to avoid too much walking (Donkey Kong 64). They were particularly useful back when there was no battery backup, since they would save players from having to replay areas every time players started again. In the Super Mario Bros. series, warp pipes are used in warp zones to transport the player from one area of the game to the other. The original Super Mario Bros. game contains a bug that allows the player to enter a Warp Zone in such a way that it will bring the palyer to the Minus World, an underwater level that loops over and over.
Warp Zones are also known as "warps" in some circles and while they're not a crucial element of the Sonic the Hedgehog games, in Sonic the Hedgehog: The Movie's "Planet Freedom" they serve as a convenient form of transportation.
Their first known appearances were 1980's Pac-Man, as areas where players can walk to the far left or far right side of the screen and they take them to the opposite side and ones that actually allow players advancement passed certain levels in Crystal Castles from 1983. In more advanced versions of the game, they are orbs, or something else closer related to modern games.
"Warp Zone" is also a term used in Unreal engine mapping (level design). A Warp Zone in this sense is a method of (somewhat) seamlessly connecting two parts of a level that are not normally connected.
Imagine two rooms, separated by empty space, with no hallway in between. Placing a doorway shape on the east wall of one room and another on the west wall of the other room, then set the doorways up as a warp zone, the player could see, move, and shoot through the doorways as if the rooms were physically right next to one another, with a real doorway in between.
Aside from making it easier to create a level in sections, this enables the level designer to do some interesting tricks with space, such as creating containers that appear bigger on the inside than the outside (à la TARDIS), or a doorway on one side of a room that leads to one on the other side of the same room.
Many space strategy games employ this idea as well, often labeling it "wormhole".