Short hopping is the act of pressing Up, X, or Y very quickly, and letting go before your character leaves the ground. This will result in a shorter than normal jump. This technique is essential in high level play, and is primarily seen in the form of SHFFLing.
Some characters are harder to short hop with, as they jump very quickly. Fox has a 4 frame time period in which you need to let go of the jump button, and is thus hard. Bowser has a 9 frame period, and is much easier to short hop with.
The height of a short hop varies between each character. This technique can also be performed in SSB.