A short hop, officially a small jump (小ジャンプ, Ko janpu) is the act of pressing the jump button and letting go before the character leaves the ground. This will result in a jump that is lower than a normal jump. This technique can be performed in all three Super Smash Bros. games.
The short hop technique is an essential move in competitive play, allowing the user to perform aerial attacks closer to the ground. This is useful for spamming aerials, especially with the SHFFL technique in Melee, as the aerials can be done closer to the ground and possibly hit targets who are still on the ground, with other techniques such as double jump cancelling allowing even lower aerials. Hitting grounded targets is great for characters whose aerials play a significant part of their metagame, such as Ness, who can perform deadly double jump cancel combos in conjunction with other moves such as up tilt in Smash 64. Characters who have strong aerial attacks such as Captain Falcon benefit from this greatly. Short hopped aerials are also very useful for aerial approaches and spacing, especially for characters such as Marth.
Jumping speed can affect the timing of the short hop. In Melee, for instance, a Fox player must release the jump button within 2 frames to short hop. Bowser, by contrast, requires only 7 frames and his short hops are considered much easier. The height of a short hop varies from character to character as well. Short hopping is also used during wavedashing, to an extreme degree.
In Brawl, it's much easier to short hop than in Melee due to the slower speed of the gameplay.
Short hopping is a bit harder to do in the Subspace Emissary, because characters jump faster.
Short hopping can also be used for Bat dropping.