Food allergies happen when the immune system treats a normally safe substance as if it were a threat. Allergic reactions can happen as soon as the food enters the body, or sometimes, in extreme cases, all it takes is a person touching or breathing the allergenic substance for a reaction to occur. Food allergies can be triggered by any amount of the allergenic food, and are a reaction of the immune system, in the same way that your body's reaction to a virus is, though the symptoms that show up may be different.
Food intolerances, on the other hand, generally take place in the metabolic or digestive systems, and happen when the body can’t digest a certain food properly. The reaction takes a little bit longer with food intolerances, because the food is first at least partially digested. Because food sensitivities have a delayed reaction, are milder than allergies, and are often caused by additives or ingredients - not a particular food itself - these sensitivities can be harder to pin down and diagnose. They also don’t happen every single time a person eats the food they’re sensitive to - they may only flare up with a large enough dose to trigger the sensitivity. Despite the difficulty of diagnosing them, food sensitivities are significantly more common than true food allergies.
Concerns about food allergies and sensitivities are the reason why it's important to introduce one food at a time to new eaters. When babies react badly to new foods, it's important to be able to pinpoint which food is causing the reaction, and what exactly the reaction is.
If you have questions about food allergies or sensitivities, you should talk to the doctor, as they are the best resource for opinions on these sorts of things.