Rabbits of the Mushroom Kingdom, much like their "Real World" counterparts, are small, furry mammals with long ears and large hind paws. They're common in the fields and wooded regions of the kingdom. These animals also feature quite prominently in several Mario games.
Nintendo Comics System
Nintendo Adventure Books
In Leaping Lizards, Morton Koopa Jr. sneaks into the palace, vandalizing it and transforming the Mushroom King into a rabbit with his magic wand. Luigi's attempt to change the king back with the same scepter only makes matters worse, changing the king from a normal bunny into a human-sized one. In a later book, Unjust Desserts, Magikoopa sneaks into the palace garden with some enemies he had shrunk by shapeshifting into a rabbit.
Super Mario 64 DS
In Super Mario 64 DS, rabbits feature much more prominently. They are found throughout the castle grounds (and their locations vary from character to character). The designated rabbits for each character have a specific color - yellow for Yoshi, pink for Mario, green for Luigi, and orange for Wario. If caught, they would unlock mini-games for the player to play in the Rec Room.
Special Glowing Rabbits need to be caught to unlock a white door in the Character Door room, behind which was a secret Power Star. They will appear only after collecting enough Power Stars. After catching eight of these rabbits, they will no longer appear.
The Korean version doesn't include rabbits because it doesn't have mini games. All the other versions do include mini games, so the rabbits do appear in the game.
Yoshi (Yellow Rabbits):
Mario (Pink Rabbits):
Luigi (Green Rabbits):
Wario (Orange Rabbits):
Super Mario 3D World
Rabbits appear again in Super Mario 3D World, where they can again be chased and caught. Larger variations also appear. When they are caught they give prizes like Power-Ups, Coins, Green Stars, or even a Mega Mushroom. Rabbits that have a green light surrounding them means they have a Green Star.