A double jump is a mid-air jump that can be used once every time the user is airborne. Just like the single jump, double jumping is achieved by hitting up on the control pad/stick or hitting the jump button.
Double jumps are a lot more varied than single jumps. Some characters twist, others flip, and there are some double jumps that do something completely different. If a character flips, his or her collision bubble moves accordingly, but this hardly has an effect on the jump. There is always the slim chance that it could narrowly avoid an attack, but very unlikely considering most hitboxes are round.
Certain characters can inflate themselves (Kirby, Jigglypuff, Dedede) or flap their wings (Meta Knight, Charizard, Pit) to jump multiple times in the air. These multi-jumps become useless (or nearly useless for some) if the character is metal.