You cannot legally drive in any state without demonstrating financial responsibility for damages or liability in the event of an accident. In most states auto insurance is mandatory as proof of this responsibility.
All states have financial responsibility laws so, in states where there is no liability insurance requirement, you need to have proof of sufficient assets to pay damages, medical bills and more if you cause an accident.
Without this proof of assets, you face legal penalties (like a suspended drivers license and car registration). Worse, in case of an accident, your responsibilities could wipe out your financial future.
If you've financed your car, your lender may require comprehensive and collision insurance as part of the loan agreement. In other words, they'll want insurance to protect their financial asset—the automobile.
You'll want to protect yourself and your assets with car insurance, as well. The insurance industry and consumer groups generally recommend a minimum of $100,000 of bodily injury protection per person and $300,000 per accident, since accidents may cost far more than the minimum limits mandated by most states.
Next steps: Understand what is covered by a basic auto insurance policy.