There are a total of 13 tamable beasts, however only 12 are available to breed.
Breeding is unlocked through the completion of Deek's side quest "Foal of the Dead" - you can buy a breeding pen spellcraft from Tomes and Scrolls in Hogsmeade.
You require a male and female of the following:
Diricawl
Fwooper
Giant Purple Toad
Graphorn
Hippogriff
Jobberknoll
Kneazle
Mooncalf
Niffler
Puffskein
Thestral
Unicorn