Double Jump is a term used for the ability to jump, and then jump again before the character lands. This is generally used as a way of getting to higher places, and assisting attack evasion.
In most cases it has no real purpose, outside of an arbitrary mechanic to design levels around.
The double jump's involvement with genres can vary. The mechanic has made its way into 3D action games such as Devil May Cry, as well as platformers like Ghosts and Goblins. A game such as Symphony of the Night builds off of the double jump by allowing the player to perform a jump kick after the move.
Triple Jumping has appeared, as well. All of the characters in Super Smash Brothers are able to do this, though their triple jumps also serve as attacks.
Most of the time, nothing is done to explain why the characters possess such an ability, beyond relying on the player's suspension of disbelief.
The first platform game to have a double jump feature is Dragon Buster, an arcade release from 1984. (Source: Wikipedia) But, well, since SotN, nearly EVERY platformer (particularly 3D ones) has had a double jump.