It can happen that a passenger falls ill during a flight. Depending on the severity of the situation, the problem will be managed differently.
If the situation is not critical, the flight attendants will handle it, often with the support of any doctors who might be among the passengers. Every aircraft carries one or more medical kits containing everything needed for first aid (for emergencies only! Don't ask flight attendants for a headache pill from the kit; they are not authorized to open it for minor issues).
In more serious cases, where a patient’s life may be in danger, the pilots will declare an emergency to Air Traffic Control (to gain priority over all other aircraft in the area) and request to land at the nearest airport equipped to handle the situation.
Many airlines use a specialized medical service that can be reached via satellite phone from anywhere on the planet. These doctors provide a legally binding opinion on the initial treatment to be administered and on whether an immediate landing is necessary. Flight attendants are specifically trained to gather and relay vital signs and medical information to these remote doctors.
What happens if a pilot feels unwell?
As mentioned [here], a single pilot is perfectly capable of landing the plane in total safety.
Furthermore, to minimize the possibility of both pilots falling ill at the same time (for example, due to food poisoning), they are required to eat different meals.
DontWorryFlyHappy!