Minimal insurance policies cover only third parties (including the insured person and third parties carried with the vehicle, but not the driver, if the two do not coincide). Also the third parties, fire and theft are common insurance policies, while the all inclusive policies (kasko policy) which include also damages of the vehicle causing the accident or the injuries. It is also common to include a renounce clause of the insurance company to compensate the damages against the insured person in some cases (usually in case of DUI or other infringement of the law by the driver).
Auto insurance is financial protection, and not just for the investment you made when you bought your car. After a really serious accident, bills for damage and injuries can easily reach into hundreds of thousands of dollars. If you happen to cause such a wreck, the victims could sue you. In the worst case scenario, assets such as your savings and home could be seized.