It's common for people to misinterpret or mislabel a food intolerance for a food allergy and to confuse the idea of a food allergy with an intolerance.
For example, recently I told a friend that a family member gets stomach aches whenever she eats certain foods. That person then said, "Ya, my nephew is allergic to peanuts."
The two things might seem related, but they're very different.
A food allergy is any immune system hyper-response to something in food – an inherent ingredient of any food item. This can be to proteins, specific chemicals the plant produces, etc. By contrast, an intolerance is when something in food causes discomfort or short-term physical ailments unrelated to immune system response.
A good way to illustrate this is with dairy products, milk in particular.
Babies can be either lactose intolerant or lactose allergic. The former is common, the latter very uncommon. A lactose intolerant baby will get colicky, throw up, or have diarrhea as common symptoms of their intolerance. A lactose allergic child will break out in hives, have serious respiratory issues, or other possibly life-threatening reactions to milk via their immune system response.
Allergic reactions generally come in one or more forms: hives, rashes, dangerous vomiting or diarrhea, itchy eyes and runny nose, swelling of tongue or throat, wheezing or trouble breathing, and in worst cases anaphylaxis (a combination of these symptoms resulting in possible death).
It's important to know the difference between an intolerance and an allergy.