Taming is the process of turning a wild, tamable animal into a domestic one by using a specific method to make it befriend you.
There are a few wild animal species which can be tamed:
A wolf will turn into a tamed wolf (also known as a dog) when given several bones. A tamed wolf will have black eyes, opposed to the red-eyed aggressive wolves. A tamed wolf will also have a red collar around its neck, which can be dyed a different color by using the dye of your choice on the wolf itself. A tamed wolf will show its health with the position of its tail, the more elevated the tail, the greater the health. It will attack any mob that you attack, acting as your side-kick. However, it won't attack creepers, and can't enter The Nether.
An ocelot can be tamed by coaxing it toward you with raw fish, and then feeding it several. This will turn the ocelot into a cat. It is important that you try to stand as still as possible while attempting to feed a wild ocelot, as they scare easily. Too much movement, and they will run away. The resulting cat that the ocelot transforms into will be one of three random breeds: tabby, tuxedo, or Siamese. Tamed cats will follow the player, but will also wander around and explore their immediate surroundings.
Horses, Donkeys and Mules
To tame a horse, donkey or mule, a player must attempt to ride the horse until it befriends you. This is accomplished by attempting to mount it several times, until it no longer bucks you off. Afterwards, the player is able to ride the animal until it despawns or dies. However, it can only be controlled via a saddle.