Pet dander is the dead skin cells that are shed by animals. When people are allergic to pets, they are typically allergic to pet dander, saliva, oil glands in the skin, or urine. Symptoms of a pet allergy include sneezing, runny nose, and itchy eyes. Some people may also develop a rash on their face, neck, and upper chest. For some people, pet dander may be an asthma trigger and they may develop wheezing, coughing, shortness of breath, and other breathing difficulties.