This is the most likely form of dental emergency and the most common mechanism is trauma.
There are no red flags to speak of with a cracked or chipped tooth. You will want to monitor the open socket for infection, so the usual symptoms of fever and pain out of proportion apply.
You need to make sure that the fragments cannot compromise the airway—remove them. You will want to position the patient face forward to allow drainage to exit the mouth rather than into the back of the throat (which makes many people nauseous).
Rinse the mouth with water as close to body temperature as possible—hot and cold are really painful on exposed roots.
Cover the exposed remainder of the tooth with dental wax. You can use clove oil to numb the area and help with the pain.
The patient will need to get to the dentist as soon as possible, but this is not an emergency unless the pain cannot be controlled or there are signs of infection.