When you tame an animal and train it to be a pet or work on a farm, that's domestication. The domestication of dogs began thousands of years ago, which is probably why they're such great companions today.
After the successful domestication of dogs, humans went on to domesticate many other animals, from horses and donkeys to cows and chickens. The term also applies to the process of breeding and growing plants for specific uses, mainly as food sources. Domestication comes from the Medieval Latin domesticare, which means "to tame" and "to dwell in a house." Despite the domestication of your donkey, she still probably shouldn't live in your house.