Capturing Animals
First you will need to make some taming collars.
Once you have those run around until you find the animals you want, you will want a male and female of any species you wish to breed.
When you find an animal you want to capture simply walk up next to it and tap it with a taming collar selected in your hotslot. You will lose one taming collar and the animal will disappear, but a pocket animal of the same type you tapped will appear in your inventory.
 Stable Setup
Now you will need the appropriate stable.
And you will need a way to keep the animals within 10 blocks of their stable, or they will become wild and disappear.
Generally fences are used for this, although you could use blocks, doors or anything else.
Fences will automatically link to a stable and form gates at each end (whether you place 1 fence or 100).
 Placing Animals
Now that you have some pocket animals in your inventory and a stable set up, you can place your animals.
Simply tap the stable with the appropriate animal selected in your hotslot.
Note that stables can only hold 3 animals and animals will not breed if the stable is full.
So if you want a baby you should only put 1 male and 1 female in a stable, to leave room for the baby.
 Caring For Animals
You will need to keep your animals happy for them to breed.
To do this you simply need to keep them fed. You can give them food by tapping the stable and moving a stack of the appropriate food in the foodstuff slot of the interface. From this interface you can also see what animals are linked to the stable and their happiness.
 Happy Animals Make...
If you did everything right (and there's room in the stable) a baby will appear.
Once the baby appears just keep the animals fed and happy a bit longer and it will grow into an adult (can be either gender).