Fast falling is the act of falling quicker than usual in mid-air. It is performed by tapping down on the control stick while falling. The character will continue to fast fall until landing on the platform or taking a hit. Gravity is ignored once a fast fall is begun; the character's downward velocity is simply set to their fast falling speed without accelerating. Fast falling is a technique commonly used to speed up a character's game, especially after a short hop. It is a part of SHFFL'ing, an advanced technique.
Stall-then-fall aerials have an effect similar to fastfalling, though the move is an attack, and most of the time cannot be stopped or slowed.
If players press down while hanging from a ledge, their character will begin to fast fall, whereas if pressed in the opposite direction of the ledge, the character will merely let go of the ledge. This also works with down + B attacks, except for ones that slow down falling or gain height. (An example is Fox's Reflector or Luigi's Luigi Cyclone.)
In Super Smash Bros. and Super Smash Bros. 4, the character will flash and a white star will appear at his or her torso when fast-falling (in Smash 4, the stars can appear in several places, but all are still located near the fast-falling character's midsection). In Super Smash Bros. Melee, a "whoosh" sound will be heard when fast falling, but there is no visual indicator, making it harder to pick up on, especially with other surrounding sound effects. There is no indicator in Super Smash Bros. Brawl whatsoever.
In Super Smash Bros. 4, characters can cancel a fastfall by using an aerial attack. This will set their falling speed to the normal maximum.
Currently, this data is sourced from experimentation documented in this this SmashBoards post, and is not yet confirmed via data extraction.