The double jump cancel, sometimes abbreviated as DJC, is a glitch-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.
Double jump canceling was almost completely removed in Super Smash Bros. Brawl, as using aerials during a midair jump with Yoshi or Ness will cause them to use the aerial whilst rising up. However, most of their special moves can still be DJC'd. Also, it is possible for Yoshi and Ness to perform a pseudo-double jump cancel on certain stages via platform cancelling, which is useful for mindgames and following up with grounded moves.
Ness
Ness' play in Smash 64 is almost completely reliant on this technique. Because his aerials provide high hitstun and damage, as well as having large combo ability (especially when coupled with other moves such as his up tilt), he can perform zero-death combos on several characters. At one point this ability was even considered broken. However, it has since lost such recognition due to Ness' short range, but nonetheless is an important part of his metagame and should still be commonly used in high level Ness play.
In Melee, Ness can still use the technique, but it is not as useful due to the nerfs he suffered in the game.
In Brawl, Ness can pull off some techniques with a DJC'd PK Fire, such as the PK Jump.
Peach
Peach can use this technique in Melee, but it is generally unused due to float cancelling being significantly quicker and more useful.
Yoshi
Yoshi's double jump cancel is extremely useful in Smash 64, allowing him to quickly throw out neutral aerial chains on floaty characters and up aerials to continue combos on fastfallers. Yoshi's DJC combos are most effective on floaty characters, especially since his up tilt cannot combo them, unlike with fastfallers. Yoshi can often finish his DJC combos with a forward aerial meteor smash on most characters.
Yoshi can perform this technique in Melee as well, though it is less effective, albeit still potent and useful in his metagame.
Yoshi can also use this technique in both Smash 64 and Melee to "counter" attacks because of his double jump's unique properties. This is referred to as a double jump cancel counter, and is often used to catch the opponent offguard with a neutral aerial.
One obvious downfall of this tactic in both games is that Yoshi loses his double jump, possibly resulting in a self-destruct when offstage. This can be extremely detrimental to him, as he lacks a third jump, and if Yoshi is hit away from the stage, Yoshi will have almost no means of recovery, outside of the directional air dodge in Melee.
Mewtwo
Mewtwo gains a significant advantage over other characters in that he can utilize the DJC. The best use of the DJC for Mewtwo is speeding up his forward aerial, though it can be used to assists his up aerial and back aerial as well (neutral aerial and down aerial take too long to be truly helped by this technique). As with all characters, Mewtwo must not perform the DJC frame perfect, as he will hit the ground before his air attack connects.
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 can cause superjumps.
Double jump cancel fast fall L-cancel
Double jump cancel fast fall L-cancel (abbreviated as "DJCFFL") is a tactic that is simply a quicker application of the double jump cancel, and is most often used by Ness, Mewtwo, and Yoshi in Melee. It allows the character in question to attack aerially even faster than with a SHFFL'd aerial.
The technique doesn't exist in Smash 64 because fastfall can't be input during an aerial attack.
Instant double jump cancel
Double Jump Cancelling works by allowing the user to perform an air attack while remaining close to the ground (most aerial attacks are performed at a short hop's distance from the ground). However, if the user performs a DJC with frame perfect precision (known as an "Instant Double Jump Cancel" or IDJC), they will actually hit the ground before the hitbox from their air attack comes out. This is true of every attack for every character who can DJC. It should be noted, however, that performing an IDJC is incredibly difficult under normal circumstances, and most often, players will be performing a DJC.
Execution
To perform this technique, the player must perform their first jump, then double jump just as they leave the ground, and press A. The player's double jump will be canceled and they will do the attack and immediately fall. The player may fast fall almost immediately, and L-cancel the aerial very quickly. The result will cause the character using this technique to attack extremely quickly.