The Carrot summons a pet Bunny when used. It grants the Pet Bunny buff that lasts until the player manually cancels it by right-clicking the buff's icon. The color of the bunny's fur is defined by the shirt color chosen for the character at the point of their creation.
It is only available to players who have purchased the Collector's Edition of Terraria — new characters created by them will spawn with it. While it can be traded, other players will not be able to use it.