If I was a passenger in another person's vehicle and there is an accident, is their insurer liable for compensating for my injuries?

In Massachusetts, if you were an injured passanger, the car that you were driving in would be responsible for provided PIP coverage (which pays the first $2000 of your medical bills). 

With regard to payment for your pain and suffering, it depends on who was at fault for the accident.  If the driver of your car caused the accident, you can make a claim against his insurance for your pain and suffering.  If, however, another car was at fault, you would have to make a claim against the other driver's insurance in the accident.