Jump-canceling may refer to two different techniques in the Super Smash Bros. games: canceling a jump by using an attack, or jumping while using the shield.
Canceling the jump with an attack
The jump-cancel occurs as a result of the pre-jump lag every character has. After the player presses the jump button, there is a period of delay ranging from 3 to 8 frames before the player's character actually leaves the ground. During this delay, since the character is grounded, it is possible to execute a few of the character's standing moves. These moves are: up smash, up special, grab (in Melee only) and any item throw, the latter which allows for jump cancel throws in Brawl, SSB4, and Ultimate.
The up smash works because of the jump delay: to perform it, the player needs to press A while holding up on the Control Stick during the jump delay, which may have been created likely to help players perform the up smash without jumping instead. The same can be said for up special moves and up smash item throws; it is unknown why grabs in Melee can be used during the jump delay, however, while other attacks like jabs and tilts cannot.
One application of the jump-cancel in Melee is the jump-canceled grab. Most characters have more range, less startup, and less lag on the standing grabs than on their dash grabs. The jump-cancel allows these characters to use their standing grab instead of their inferior dash grabs while running.
This tactic is also what allows one to do a running up smash by dashing and jump-canceling into an up smash. The player can also jump-cancel an item throw to use the faster normal item throw instead of a higher-ranged running one.
This technique is more difficult to perform in Ultimate due to the universal jumpsquat change to 3 frames and short hop aerial macro. However, the newly introduced ability to cancel dashes into any move more than makes up for it.
Leaving the shield with a jump
Another usage of the term "jump-cancel" refers to the cancellation of a special animation by jumping, mainly shielding; jumping out of shield is possible in every game with every character except for Yoshi in Melee and Brawl. Additionally, Fox and Falco can do the same in Melee with their shine. The animation achieved by turning around while dashing (after leaving the initial dash animation) can also be jump-canceled.
It is possible to jump-cancel a shield into a jump-canceled attack (most commonly Bowser and Link's up special, or Fox and Pikachu's up smash); in this instance, only one use of the word "jump-cancel" is commonly used: "Link used a jump-canceled up-B from his shield." This is less useful in Ultimate, however, because up specials and up smashes can be instantly performed while shielding without jumping. One can also jump-cancel the shield into a wavedash, because the wavedash starts with a jump.