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
Cancel is a general term in any fighting game and can mean many things, depending on the context. Usually it refers to canceling a normal into a special in the Street Fighter series, denoted by "xx" in combo notation. This is also known as a "two in one", as you perform the motion for the special immediately after inputting the normal. An example of a cancel with Ryu is: low forward xx fireball, which means you would input down+forward kick, then immediately move the stick to down forward, forward, and press punch.
Some moves are cancelable and some are not; refer to each individual game for details.