How do you breed animals?

Feed both animals their specific food at same time, while they are beside each other. That will make them have hearts above their heads and breed with each other.

• Sheep: wheat
• Cows: wheat
• Pigs: carrots or potatoes
• Chickens: seeds, any type (wheat, pumpkin, melon, or even Nether wart)
• Horses: Golden Apple or Golden Carrot
• Tame wolves: bones (to tame). Rotten flesh and any form of meat, cooked or raw
• Tame cats/ocelot: raw fish (sneak, holding fish so it walks up to you, then tame)
• Turtles: seagrass (can be obtained with shears)
• Dolphins: raw cod
• Polar bears: fish (any kind)
• Pandas: bamboo
• Parrots: seeds

Take a look at Breeding on the Minecraft Wiki.

Basically, you must feed them both one wheat. Then they would go towards each other and hopfully have a baby.

Here is actually a video of the 'action':

The only thing not shown in this video is the fact that the new 'babies' are actually a baby form of the animal, they are smaller and much cuter :)

Pigs, Sheep, Cows and Chickens can all be "breed" by feeding them wheat. (To feed an animal wheat, carry wheat, and right click the animal.) Once an animal "eats" the wheat, it gets hearts above it. Once two animals get close to each other with hearts above them, they create a tiny animal. Each animal can reproduce once per minecraft day (or so.)

Chickens can reproduce in another way. You can take an egg, and throw it on the ground. About 1 in 8 times, a chick will hatch from the egg. This is the only way to get a passive mob from the normal world, to the nether or the end.