Food allergies can be fatal. When a patron discloses they have a food allergy, listen deeply and plan your actions carefully. Sometimes the diner may disclose a dining preference versus an allergy. It's wise to deal with the preference in the same way. To demonstrate, I'll walk you through an example. If a diner states they have a shellfish allergy:
Ask them to define the allergy. The patron is the authority on their allergy and how their body responds. If they say they have a shellfish allergy, ask "Is it all fish? Which shellfish?" They may be able to consume clams but not lobster or crab. The patron will be pleased you are taking such a strong interest, so don't be afraid to ask and clarify.
Notify the Kitchen. Describe to the kitchen exactly what the patron said. The chef may send you back to ask more questions, if s/he does - do so and follow up thoroughly.
The patron may use terminology differently than you do. Your job is to make the patron feel comfortable and to provide a positive experience. If the patron states they are vegan than asks how the fish is prepared, don't judge or correct the patron on the definition of vegan, ask questions to clarify. Try, "I just want to understand, you are vegan, which fishes to you enjoy? Are dairy products acceptable, eggs?" Clarifying understandings will go a long way in creating a positive dining experience for the patron.
A Clearly Defined Menu helps. There are universal symbols for food allergens as defined by the International Association of Food Protection but, again, everyone will likely not be familiar with all of them. Clearly written, engaging descriptions of the items in each selection will help a lot. Highlighting GF (gluten free) V (vegetarian) and VG (Vegan) next to selections and include foot notes at the bottom. There is no substitution for knowing the menu and clarifying with the kitchen.
If there is a problem know your restaurants policy or ask your manager, but most restaurants do not disseminate any type of medications due to potential risk. If a patron asks for an aspirin, you have no idea if they can tolerate aspirin or how it might interact with anything else they are taking. Many patrons with severe allergies will carry medicines with them in case there is a challenge. You may come across the unfortunate experience where the patron does not have medicine or is having an allergic reaction for the very first time. If this is the case, direct them to the nearest emergency room, pharmacy or if the situation warrants call 911.
In summary, know your menu, let the patron define their allergy or dining preference in detail, confirm with the kitchen, follow up to make sure dining requests were accommodated before delivering food. If there is a problem, take it very, very seriously and respond accordingly.
In some states allergen training courses are mandatory. Even if you don't happen to be in one of those states it can be good practice as an employee or smart business practice if you're an owner/operator.
**As a note it is not always possible to accommodate every diners allergies in every dish. Being able to communicate what you can and cannot do safely to accommodate them is just as important.