Taming is the process of turning a wild, tamable animal into a domestic one by using a specific method to make it befriend you. A wolf will turn into a tamed wolf (better known as a dog) when given several bones. The same process is needed to get an ocelot to befriend you, turning it into a cat, with the exception of the bones being raw fish.
A tamed wolf / dog will show its health with the position of its tail, the more elevated the tail, the greater the health. A tamed wolf will attack any mob that you attack, acting as your side-kick. However, it won't attack creepers, and can't enter the nether.
For horses, however, a different method is necessary. The player must attempt to ride the horse until it befriends you (this will look the same as if taming a wolf or ocelot, much as if the animal has entered "love mode"). Afterwards, the player is able to ride the horse for an infinite period of time, but can only be controlled via a saddle.