This video demonstrates how to perform cancels.
Cancel is a common term across fighting games. Canceling refers to interrupting an animation that is in progress with another animation. So you may see half of the first animation, up to where it is interrupted, and all of the second one. The most common type of cancel is the normal to special cancel, also known of as the 2in1 cancel, because it's like having two moves in one, since you need to commit to using both moves at once. Cancels are denoted by "xx" in combo notation. Performing a cancel in most games means pressing the button for the normal attack, then inputting the motion before the move hits, then pressing the button to activate the special move right when it connects, during the hit freeze.
The 2in1 Cancel originates in Street Fighter II. The Original Street Fighter had extremely difficult to use special moves, so in the sequel a number of methods were introduced to make them easier to perform. The window for each directional input was increased, so they could be input more slowly and sloppily. The system was changed to react to the release of buttons (negative edge), instead of just the press of buttons for triggering special moves. And a 5 frame cancel window was added to the beginning of every normal move (this window varies slightly by character in Super Street Fighter II Turbo), known of as the kara-cancel window (in english, empty-cancel). A possible intention of this window was to make it so if people did the special move motion, but pressed the attack button before the motion was complete, they would still have a chance to cancel the normal into the special move when they released the button, instead of being locked out by the normal move in progress.
This kara-cancel window was implemented completely intentionally as a way to make special moves easier to perform, but it has an unintended side-effect: whenever a move makes contact with an opponent, there is an effect called Hit Freeze. This effect freezes both characters temporarily for a short duration of time (roughly 10 frames) on the same frame where they made contact. For moves with 5 frames of startup or less, this means that the hitfreeze will freeze the kara-cancel window, meaning the window stays open for the entire duration of the hitfreeze. This has two effects, first of which is that it makes it possible to cancel into special moves after hitting your opponent, creating a combo! The second is that this hitfreeze makes the cancel window much wider than normal, making it very easy to cancel into special moves.
After this effect was discovered, Capcom decided to leave it in the game, and include it in future games, except modified so the kara-cancel window was much much smaller (1 frame instead of 5), and so the special cancel window was the hitfreeze itself. This also gave them the freedom to manually choose which moves could cancel into special moves, though they tended to stick close to what moves were cancellable in Street Fighter II. Other games have since implemented this system, and expanded upon it, allowing specials to be canceled into supers, normals to chain together into other normals, and canceling moves entirely by spending meter.
Cancels enable new combos because by interrupting the move rather than letting it play out completely, you can take advantage of the entire duration of the hitstun dealt by the attack rather than just the frame advantage left over at the end of the attack. Normally an attack's frame advantage is the hitstun left over after the attack has completely recovered. The hitstun animation might be 19 frames long, but after 15 frames of recovery on the attacker's animation, that only leaves +4 frames of hitstun, which is too short for a 17 frame startup special move to come out. By canceling the attack, the recovery portion of the animation doesn't need to play, so the 17 frame startup fits in easily. You can calculate the frame advantage of a cancel by adding together the active frames, recovery frames, and frame advantage of the move. You can then check if this combos by looking for special moves with less startup than the hitstun number you just calculated. Moves with short hitstun, like light attacks, won't always combo into special moves with longer startup, even though light attacks generally cancel.
There are many types of cancels in fighting games. Kara-Cancels let you cancel a move during its startup. Feint cancels let you cancel a move on whiff (The Street Fighter series lets fireball specials cancel into super with a feint cancel window). Jump and Super Jump cancels let moves cancel into jump, sometimes only on hit, not block. Airdash cancels allow air moves to cancel into airdash. Moves that involve you holding a stance before they become active frequently let you cancel the stance into an idle standing pose. Many Anime style fighters, such as Guilty Gear allow moves to cancel later than the hitstop, during the recovery, such as to allow chain moves to set up frame traps on block. Landing from a jump-in attack is technically canceling your jump-in by hitting the ground. Some combo video makers get hit by attacks intentionally, so the hitstun can cancel them out of an attack that otherwise cannot be canceled. In real matches, this trick is sometimes used to trade hits with an opponent when your attack will deal significantly more hitstun than theirs, letting you get a combo off the trade.