What makes an animal possible to domesticate?
The world's foremost expert on domestication is arguably UCLA Professor Jared Diamond, who is best known for his popular science books like the awesomely titled Guns, Germs, and Steel. In the book, he lays out six basic criteria that an animal has to meet in order to be possible to domesticate. Let's run through them briefly:
The animal needs to be able to eat a lot of different thing and be willing to live off the scraps of humans. If the animals are able to eat stuff humans can't, such as grass, then even better. This also makes strict carnivores somewhat more difficult to domesticate than other animals, as it commits humans to providing a ready food source of other animals for them to eat.
The animals need to grow up fast, or at least faster than humans. There's not much point in trying to domesticate extremely long-lived species like elephants or tortoises, as it can take several years before they're even remotely useful, and their long life cycle limits how quickly their numbers can be replenished.
The animals must be willing to breed in the close quarters of captivity. Any creature that demands a lot of open territory in order to breed - pandas and antelopes are good examples of this - are terrible domestication candidates.
The animals have to be naturally pleasant. An unpredictable or ill-tempered beast is just going to be dangerous to attempt to keep enclosed in a small area. It's possible to meet some animals halfway on this - for instance, the American bison can be kept in huge enclosures on ranches - but that's as close as we can get to full-on domestication for species like that.
It isn't just pleasantness - the animals need to be calm as well. Skittish or flighty animals will constantly attempt to escape, and it can be almost impossible to control them even if escape is impossible. This is what seems to have kept foxes from being successfully domesticated, as they're far more skittish than dogs and wolves.
The animals need to be willing to recognize humans as their new masters, which means they must have a flexible social hierarchy.
The remarkable thing about all this is that some animals that seem almost indistinguishable from their domesticated counterparts can prove to be incurably wild. The difference between foxes and dogs doesn't seem like very much, domestic pigs seem superficially very similar to the impossibly wild peccaries and warthogs, and zebras can even interbreed with the fully domesticated horses and donkeys...and yet outside a couple isolated cases of taming zebras, the species has proven completely impossible to domesticate.