Breeding is a feature which was introduced in Beta 1.9 Prerelease 2. If a player holds a certain food and right-clicks an animal, it enters what is known as love mode. By giving two of the same animal the specific material and insuring that they are close together, the animals will produce a baby animal. Animals in love mode will be attracted to each other, but love mode only lasts for a small period of time (roughly one minute).
Foods correspond with certain animals. Wheat for sheep, cows and mooshrooms, carrots for pigs, seeds (or Nether Wart) for chickens, fish for ocelots and any kind of meat for wolves. (Note that the wolves need to be tamed)
Animals cannot enter love mode if they recently had a baby. Breeding a cow and a mooshroom will create a normal cow. Chickens can not only be bred with seeds, but in an artificial way of throwing (right-clicking) eggs. They may also be spawned by a dispenser throwing eggs. Chicken eggs have a 1/8 chance of hatching into a chicken. By means of breeding can be useful for farming. Sheep can have different colored babies depending on the parents' color if two sheep with the same color make a baby of that color.
Villagers can also breed and make baby villagers, but it only happens naturally and cannot be forced unless in creative mode, in which case the player can use a spawn egg on a normal villager. Villager breeding is dependent on the number of occupied structures there are within the village's perimeter. Despite making more sense to leave tracking to the buildings as a structure, the tracking is set on the number of doors compared to the amount of villagers currently living. Villager breeding is extremely rare, however as many villagers need to be deceased in order for it to occur, leaving them extremely weak with only 33% of villagers remaining.
In Beta 1.9 Prerelease 3, Notch added baby animals. The baby animals are smaller variations of their parents. They have smaller bodies, normal sized heads (same as the adults), high pitched cries and walk faster. They currently don't drop anything. The infant that is born will usually follow the parents. The baby animals grow into a fully-sized animal after one day in-game (20 real minutes).