Do all states require car insurance?
No, all states do not require car insurance, but all require financial responsibility to operate a vehicle on the roadway.
Every state requires that you meet financial responsibility requirements through insurance, a bond or some other approved means that show you are able to pay if you cause damages to another person or property in an automobile accident. (See "Got $30,000 and hate car insurers?")
Each state renews its laws annually, so some states that had no insurance requirements in the past now do. New Hampshire probably has the least amount of requirements -- and it still requires that you immediately show proof of financial responsibility if you've been involved in a car accident.
If you choose to buy insurance, as most drivers do, all 50 states have different minimum insurance requirements.
Almost every state requires you to have bodily injury liability insurance to pay for the treatment of anyone you injure; other states operating under no-fault laws will you require that you carry personal injury protection to pay for your own injuries. You usually will be required to buy property damage liability insurance to repair the vehicles of anyone you hit.