Moonwalking is the act of sliding backwards during the dash animation. This action can be done in both Melee and Brawl but more commonly used in competitive play in melee.
Every character can moonwalk, but the action to moonwalk for a character is different depending on what character you are.
The pros of moonwalking are that it is easier to edgeguard mainly because if you moonwalk all the way to the edge of the stage and not jump, your character will slide off the platform and grab the ledge.