Can Cats Get Viruses?

Yes they can, they are called FVRCP and Feline Leukemia. FVRCP can cause upper respiratory-tract infections, easily transmitted from one cat to another. Kittens are most effected, and these diseases can be dangerous if not protected by vaccinations.
Feline Leukemia is generally fatal, resulting in possible cancerous conditions such as leukemia, to secondary infections caused by the destruction of the immune system.
Fortunately, vaccinating your cat annually will protect them from these viruses.