Breeding allows you to duplicate animals. Cows, Mooshrooms, Chickens, Horses, Wolfs, Ocelots, Pigs and Sheep can breed.
What you need
You need at least two of the animals food items as well as two animals of the same type. Cows, Mooshrooms and Sheep have Wheat as there food, Pigs have Carrots, Horses have Golden Apples/Golden Carrots, Wolfs have meat, Ocelots have Raw Fish and Chickens have Seeds. It is best to create a pen to keep your animals in, so Fence is recommended.
You can make animals follow you by holding their food item and moving close to them. They will follow you as long as your holding their food item. You can then lure them into your pen.
To make 2 animals breed, right click on each of them while holding their food item. They will have hearts float from them when you have used the food item on them. Both animals will run towards each other and a new one will spawn. You can repeat the process as long as you have more food. Animals require time before they can breed again.