The double jump cancel (sometimes abbreviated as DJC) is a tactic used by Ness, Mewtwo, Peach and Yoshi, who can attack immediately after their double jump to cancel their upward momentum and land on the ground soon after their attack. This tactic is used to perform aerial attacks incredibly quickly, thus allowing aerial combos and KO's to be performed more quickly.
But other characters who can cancel their double jump use this tactic to cover more horizontal distance and reach enemies who are further away. Mewtwo players generally only double jump cancel when they want to attack opponents who are out their range.
For Peach, this tactic is usable, but generally unused due to her slower aerials and superior float cancels. The pro peach user XIF is known for using multiple djc's neutral air.
For Yoshi, he can use this to "Counter" attacks because of his double jump's unique properties.
One obvious downfall of this tactic is that the character loses their double jump, possibly resulting in a self-destruct when away from an edge. This can be extremely detrimental to Yoshi, as he lacks a third jump, and if Yoshi is hit away from the stage, Yoshi will have almost no means of recovery.
Double jump canceling has almost been completely removed in Super Smash Bros. Brawl, some specials are the only moves that can cancel character's double jumps.
DJC physics
Since the affected characters' double jumps are of the "delayed" type (where they do not rise immediately), interrupting the action with an attack will cause the scheduled upwards force to not take place.
A similar mechanic causes some kinds of superjumps.