A double jump is a maneuver in some video games wherein the character using this ability performs a second new leap in the air, jumping as they would if they had been standing on solid ground. It is usually found in platformers, either as a built-in and mandatory maneuver to use1, or as an upgrade obtainable as the game continues2, although it's also possible to find in some arcade, action or reflex games3. Some claim that having the ability to double jump in real life would be "pretty wicked", while others recognize that the ability to keep your shins intact is a little more precious to them.
Typically, the mechanics of a double jump completely overwrite all vertical movement and momentum, propelling the character off from the air beneath them exactly as they would bounce from the ground. Horizontal movement can be either unaffected, allowing for more complex midair navigation techniques, or it might kill your horizontal speed as well, making their usage a much more strategic affair. Some games even feature a "recharger" item, which will restore your ability to jump again in midair—expect many evil levels based on this.
If the double jumps are an unlockable upgrade, then the well-designed adventure game will usually feature lots of high walls or just out of reach distances, for which it is obvious that you'll be coming back later.
Of specific mention, for the non-platfomers, is the use of double jumps (and, indeed, any aerial movement tactics) in fighting games. Often, the free range of motion allowed by the ability to move quickly or jump in a fighting game can give some characters an edge, in terms of being to to jump in, get some hits, and move back out. Maneuverability is everything, especially in the 3D versions of the games, so something like a double jump to utilise the playing field can be a real game-changer.
Other related video game aerial maneuvering systems include: wall jumping, the parkour-like ability to cling to and jump from walls, often endlessly if presented with a vertical hallway; gliding, which involved the character making a slow descent downwards; debug modes, which usually involve some sort of mechanism for getting around a level easily; and the Triple Jump Moonsault.
1 Some games with a second jump as a mandatory game mechanic:
2 Some games with a double jump upgrade, for game progression or otherwise:
3 Some non-platformers (that is, games that do not feature platforming as a main element) that make use of the double jump:
TVTropes maintains a longer and more comprehensive list here.