A malfunction is any parachute deployment that does not result in a canopy that can be landed safely. The category covers everything from a canopy that fails to open to one that opens but cannot be controlled.
Broad types
Total malfunctions are those where nothing comes out of the container at all. Partial malfunctions are where something deploys but does not produce a flyable canopy — line twists that cannot be cleared, a tangled canopy, a spinning canopy, or a torn one.
Some situations are recoverable and some are not, and telling them apart quickly is a core part of skydiving training.
This is a definition, not a procedure
Deliberately, nothing here describes how to respond to any of these. Emergency procedures are trained on the ground, rehearsed physically, refreshed for currency, and specific to the equipment you are wearing. Reading about them is not preparation, and improvising from a description is how people get hurt.
If you are training, your instructor is the source. If you are returning after a break, a currency briefing at the drop zone is the source.
These pages explain what things are so the language makes sense. They are not instruction and cannot substitute for it — skydiving is taught in person by rated instructors at a drop zone, and anything involving equipment handling or emergency procedures belongs there rather than on a web page.