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.
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.
State laws and minimum insurance requirements vary greatly, and that has a huge effect on what you pay. In some states, drivers can't register a car without showing proof that they have liability insurance, while other states only require proof of insurance when upon request -- basically meaning when a driver has an accident or gets a ticket.
If traveling out of state, take your physical auto insurance card (or other proof of financial responsibility) and review your policy for any out-of-state policy limitations (such as only the named insured being covered) before hitting the road.
Most policies will give you the same liability limits you currently carry while you’re out of state unless these do not meet the other state’s minimum limits. If that is the case, typically your policy will automatically raise the limits to that state’s limits if you are in an accident.
Two states, Virginia and South Carolina, allow drivers to pay a fee to register a car without insurance, though the fee does not exempt them from financial responsibility requirements. (See "Drive uninsured, legally, for $500.")