You can use it to breed Pigs. If you tame two pigs and you feed them at the same moment (appears hearts around the animal) Pigs mating and will appear a new little Pig.
Carrot farming takes 8 stages to grow. But only 4 stages are visible for the player. Every two stages have the same texture, except that growth stage 7 keeps the same appearance as stages 5-6, so that only stage 8 has the final, 'mature' appearance. Mature carrot crops drop 1 to 3 carrots.