Dogs and cats can develop allergies to the same things to which humans are allergic: "natural" or environmental allergens (grasses, trees, pollens, dust, etc) and food. Other allergic reactions exist, such as contact allergies and anaphylactic reactions, the latter being the most serious (e.g., allergies to bee stings and peanuts).
Atopy or atopic dermatitis is an allergy to an environmental allergy (inhaled and/or absorbed through skin).
The vast majority of dogs and cats with allergic skin disease are itchy. Dogs and cats might chew and/or lick their paws more than usual, scratch their armpits and groin regions, have peri-anal itchiness, scratch their neck and ears more frequently, and scratch their faces. The vast majority of ear infections can be explained by a primary allergy! If your dog or cat "always has ear (and/or skin) infections," your vet should be talking to you about primary allergies likely being at the root of the problem.
In simple terms, an allergy means that the body is not recognizing an allergen and is trying to get rid of it. So rubbing your eyes and sneezing because of allergies is a "natural" way for the body to get rid of allergies. The problem arises when the symptoms are excessive for the inciting allergen, meaning that the sneezing and scratching are so excessive that the symptoms are much more harmful than the allergens. Same thing goes for pets. Mild scratching is perfectly acceptable, but itchiness that causes a dog to scratch herself until she bleeds is evidently not acceptable. The allergy is more harmful than medication to control it.
Imagine the suffering from being so itchy. Think: allergies!