First it is important that you make sure any vehicle you drive is insured. Even if you are not the owner you can be cited by law enforcement if found driving a car without insurance on it and that would extend to you as a driver.
If you do drive a car that is uninsured and are in an accident then unfortunately if you are at fault then you will not be personally responsible for the damages you caused to that vehicle and to others. Though you have your own insurance, physical damage coverages of Collision and Comprehensive do not extend to friend's cars that you may drive. Your Liability coverages may but only as secondary or excess coverage and the car owner's coverage would be primary. If the car owner is without insurance then it is doubtful that your car insurance (Liability coverages) would step up to be primary though you could check with your insurer to find out for certain.
So you would be responsible for the damages you caused to your friend's car and to others you may have caused property and/or bodily injury to. The car owner may also have vicarious liability for your actions so both may be held responsible depending upon state laws.
If the accident was not your fault then in many states even if you were driving without insurance you could place claims for the damages with the at-fault party's Liability coverages.