When someone who has always owned a cat suddenly develops a cat allergy, this is medically known as an "allergic reaction" or "hypersensitivity reaction." It refers to the body's immune system becoming overly sensitive to some harmless substances (such as pollen, animal fur, etc.), triggering an immune response that can then harm the body.
The common skin rash accompanied by itching that people experience is a type of allergic reaction. Some people are born allergic to certain substances but do not experience allergic symptoms upon initial exposure; others may not be allergic to a substance initially, but develop an allergic reaction upon re-exposure due to changes in their immune system. 
One factor is related to an individual's constitution; some people are more prone to reacting to pet hair and dander. Individuals with atopic constitutions can be sensitized upon first contact with the antigen, but without developing a clinical reaction. This explains why some people haven't previously experienced allergies to pet hair.
Because the body's immune status is constantly changing, a sensitized organism can react when it comes into contact with the same antigen again. The timing of this reaction is unpredictable; it can occur as quickly as within seconds of re-exposure, or as slowly as several days, months, or even longer.
Pet hair and dander are inherently allergens, and if pets aren't cleaned regularly, mold and mites on their fur can also be significant allergens. Pet owners are advised to regularly clean and disinfect their pets to reduce the likelihood of allergies.
