Most states require auto insurance. Technically, you prove your financial ability to pay a specified level of damages when registering a vehicle or renewing license plates. The only way for most people to satisfy this requirement is to have car insurance for the minimum amount of coverage.
If you have a car loan or if you lease your vehicle, the loan company or leasing company will also require that you have insurance. You name the company as a "loss payee" on your policy. If the vehicle is damaged, any insurance payment will go to the company, and it uses the money to either to repair the vehicle or pay off the loan balance.