A yellow ribbon of magic spins around Zelda, causing her to disappear. After that, she will reappear in the direction held on the control stick (or directly above her original location if the control stick is not moved). The initial ribbon spinning is capable of damaging opponents that touch it.
In Super Smash Bros. Brawl, Zelda teleports a slightly shorter distance than she did in Melee, but also creates a small explosion (similar to the one made by the initial part of Sheik's Vanish) upon reappearing, dealing some damage and knockback. Farore's Wind is hard to sweet-spot the ledge with.
It is based on a spell with the same name from The Legend of Zelda: Ocarina of Time. Link could use it to mark a place in a Dungeon and then return to that warp point at a later time. It is the spell which is representative of Farore, the Goddess of Courage.
 Description from the Melee Instruction Booklet
 Description from the Brawl Instruction Booklet
Spin in midair and teleport far off in the direction you tilt the Control Stick.