If someone other than a family member is involved in an accident in your vehicle and is at fault whose insurance is responsible?

Barring any exclusions in the policy, the insurance 'stays with the car' so your insurance will pay for the damage to the innocent persons vehicle (under your liablity coverage). If you have collison coverage on your vehicle/policy it too will pay to repair your vehicle less the deductible. If there is no collision coverage on your vehicle and the driver has a policy with collision coverage the drivers collision coverage may step in and repair your vehicle, but ONLY if you don't have collision coverage.