Carrot is a food item that can be planted and consumed: it restores 4 hunger points.
Carrots can be used to breed and attract Pigs.
Carrots are also a rare drop from zombies, and are found as crops in Villager farms.
Along with potatoes, carrots require farmland in order to grow:
A carrot item is planted directly in farmland, and upon maturing will yield 1 - 4 identical carrot items.
A carrot can be used to craft a Golden Carrot, which can be used to make Potions of Night Vision and breed Horses.
The carrot on a Stick can be used to steer pigs, although since 1.6, Horses are a better alternative for riding.