Full coverage encompasses comprehensive and collision coverages typically. Generally insurance covers a vehicle first and driver second. So if someone that you gave permission to drive your vehicle crashes generally your insurance would pay for the damages with the coverages you have. Sometimes insurance companies will exclude certain drivers for various reasons so it is always wise to contact your insurance company to make certain your friend or others will be covered if they drive your car.
It is the law in most all states that drivers have liability insurance so even if the car is insured the driver needs to have their own insurance so if they are stopped they will not get a citation for driving without insurance.