A Kara cancel is 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). Only certain moves can be kara canceled (usually normal attacks), and what they can be canceled into varies (usually specials, supers).
Kara cancels were put in by developers as a way to make inputs a bit more lenient. For example, a player mistiming a as ~. The window to kara cancel these moves usually isn't very long (usually 2-3 frames).
At this point you may wonder why would you want to do this? 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.
- GBVS Katalina kara special: c.M > M > 623M
- Katalina's auto combo moves her forward during the first few frames. By canceling c.M > M into 623M, she can give it more range for free, allowing her to combo into 623M from further distances.
- 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.
- GGST Potemkin Kara Buster: 6K > Specials
- Potemkin's 6K moves him forward during the startup. By kara canceling the 6K into various specials (especially Potemkin Buster), you are giving Potemkin's special attacks more range.
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.