Falling speed is the rate at which a character can move downward in mid-air. A character's falling speed can greatly impact the fighting style - for instance, Fox, a fast-faller, uses his aerial speed to perform SHFFL'd attacks very quickly, whereas Jigglypuff, a floaty character, uses its low falling speed to attack multiple times in mid-air.
Additionally, falling speed directly affects the vertical component of attacks' knockback in combination with weight. Attacks of upward trajectory knock back slow-falling, floaty characters more than they knock back fast-fallers. However, a floaty heavyweight will be harder to KO vertically than a floaty lightweight. Though both falling speed and weight affect knockback, they are independent of each other - a fast-falling character is not necessarily heavy, and vice versa. Samus and Fox are notable examples of this.
The items Bunny Hood and Metal Box increase the user's falling speed. Some stages can also affect a character's falling speed, such as the flying-type portion of Pokémon Stadium 2 and the falling portion of Summit. All characters can also fast fall at any time to increase falling speed.
Categories of falling speed
Characters are generally put into one of two groups of falling speed: floaty and fast-faller, each of which have certain advantages and disadvantages depending on playstyle.
Fast-faller is a name that refers to characters with a fast falling speed. Fox and Captain Falcon are prominent examples of characters with very high falling speeds in the Super Smash Bros. series, with Fox being unusually light for his falling speed.
Fast-fallers enjoy increased speed in the air and faster SHFFL'd approaches; however, they suffer from shorter recoveries. They are also susceptible to different combos from floaty characters. For instance, Captain Falcon can chain grab floaty characters with his down throw, but cannot do so against fast-fallers. On the other hand, he can tech-chase fast-fallers (or chain grab them given the right percents) with up throw, which he cannot do against floaty characters. Generally, fast fallers are more vulnerable to chain grabs and combos (although weight and character size are other factors as well); such as Marth's up throw chain grab on the space animals in Melee that he can finish with a forward smash. (although notably this chaingrab does not work on fast-faller Captain Falcon as he is too heavy)
Floaty is a name that describes characters with a slow falling speed. Jigglypuff and Samus are prominent examples of very floaty characters in the Super Smash Bros. series; Samus being floaty, however, is unusual for a heavyweight.
Floaty characters have great benefits in terms of performing aerial attacks, since one would able to perform multiple aerial attacks in succession (such as Jigglypuff's slow falling speed and multiple mid-air jumps allowing it to do the wall of pain) and easier recoveries due to slow falling speeds allowing easier maneuvering. Downsides include vulnerability to being juggled, having easily predictable recoveries, or being KO'd vertically easily. On the flip side, floaty characters tend to be less vulnerable to chain grabs and combos.