A Kara cancel is the canceling the startup of a move into another move. This name comes from the Japanese word 空 (kara), which means empty or sky. You are canceling the startup of the first move such that it never comes out (and thus empty).
At this point you may wonder why can you cancel the startup of a move? Developers put this into the game to act as a way to make inputs a bit more lenient. For example, a player mistiming a Template:236 as Template:236~. The window to kara cancel these moves usually isn't very long (usually 2-3 frames).
Competitive players have found more advanced applications for this by finding specific moves with certain properties that would benefit another move.
- BBCF Tsubaki kara throw: 6C > Throw
- Tsubaki's 6C moves her forward during the startup. By kara canceling the 6C into a throw, you are giving Tsubaki's throw more range.
- P4AU Yosuke Glide: Air dash > j.2AB > j.AC
- Yosuke's j.2AB makes him fall towards the ground during the first few frames. By air dashing, then j.2AB, then canceling it into an air turn (j.AC), he basically gets an air dash that goes diagonally towards the ground. This gives him an additional movement option that lets him attack from new angles.
- GGXRD Ky momentum stop: Air dash > Ride the Lightning > YRC
- Ky's Air Ride the Lightning stops his momentum during startup. By kara canceling the Ride the Lightning into a YRC, Ky can stop his forward momentum and fall straight down. Ky players usually use this with a meaty j.D to create crossups while staying close enough to combo afterwards.
All these examples are character specific, but they all use the same principals of kara canceling - taking the properties of one move and applying them to another.