An allergist is a doctor whose specialty is allergies. If you suspect that your cat is making you sneeze uncontrollably, you might want to see an allergist.
Allergists can diagnose allergies, perform tests to find out exactly what patients are allergic to, and prescribe medication to treat the symptoms of those allergies. It takes extra years of study to specialize in this field and become an allergist. The word comes from allergy, by way of the German Allergie, coined by an Austrian doctor from the Greek roots allos, "other, different, or strange," and ergon, "activity."