An illness that can be spread between animals and humans is a zoonosis. Rabies is one example of a zoonosis that's transmitted when a rabid animal bites a person.
Some kinds of zoonoses can be directly transferred from animal to human, while others go through a third animal that acts as a carrier but shows no symptoms. If a human transmits a disease to an animal, it's known as "reverse zoonosis." The list of zoonoses includes cat scratch disease, anthrax, avian flu, and Rocky Mountain spotted fever. The word comes from the Greek roots zōon, "animal," and nosos, "disease."