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. The Donkey Kong Country series features a roll which serves a similar purpose.
Triple Jumping has appeared, as well. All of the characters in Super Smash Brothers are able to do this, though their 'third' jumps' primary function is as an attack.
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. At most, a game may offer some sort of “magical” explanation, such as in the Devil May Cry games, where the main character can make a magical platform appear in the air, which he then jumps off of to achieve the double jump.