Allergies or hypersensitive reactions, present in ways depending on a number of factors. For example, what the person is allergic to and how their body responds when exposed to that allergen. The most common groups of allergens include foods (peanuts, shellfish, eggs, milk), medications (eg. antibiotics), vaccines (eg. typhoid, pertussis), venoms (eg. bee stings, wasps), cosmetics, and other environmental chemicals such as those found in plants, dusts, pollen, mold, pet dander or saliva, etc.
The most common type of allergic reaction is Type 1 (anaphylactic) reactions. The symptoms usually come on quickly where the person may experience eye, nose, and mouth/throat itching or swelling, hives or welts, and mucus production which can affect their airways and breathing. Anaphylactic shock is a medical emergency and most people carry an Epipen with them in case of exposure (quick administration of epinephrine which helps to open their airway).
A true allergic reaction generally follows the same symptom picture no matter if from food or from something in the environment. Signs to watch for include red patches or welts on the skin, swelling or itchy lips or eyes, runny nose, fatigue, and difficulty breathing. The chart available here shows the difference between colds, flu's and allergies.