State laws differ in regards to if only the car need to be insured or if both the vehicle and driver need to have insurance on them. Some states require you to have insurance on you in order to apply for a license or renew your driver's license. In other states the driver does not necessarily need insurance in their name if they do not own a car but only drive other people's cars that already have insurance on them. In most states to register a car you must show proof of insurance.
If you do not own a car, and do not have your own insurance, then you will need to make sure that the auto insurance of those that let you borrow their vehicles will cover you if you are in an accident. For example, if someone in your household has a car that you use then you would probably be required to be on that person's insurance policy in order to be covered when operating their car.
If you are using cars that are not from within your household then the family member or friend which you borrow the car from should check with their insurance carrier to make certain that you will be covered when driving their car, even though you are not listed on the policy.
To find out if your state requires you to have insurance on the car or the driver check with your local Department of Motor Vehicles.
If you are required to have insurance in your state to carry a license but do not own a vehicle then you may need to obtain your own non-owners insurance policy. Follow this link for a cheap car insurance quote.