The reason we are required to have auto insurance by law is to cover any damages that may occur when you cause an accident.
Without auto insurance then you would personally be liable to cover the cost of any damages or injuries to someone or someone's vehicle if you cause the car accident.
On a used car or a car that you own outright you are required in most states to have at least Liability auto insurance to cover any injuries and damages to someone's vehicle.
If you have a vehicle with an auto loan then you are required by the lender to have full coverage or comprehensive auto insurance until the auto loan or car loan is paid off.
Basically having auto insurance by law makes you legally responsible and insures that you have the money to pay someone in the event you cause an accident and injure, kill someone or at least damage the persons vehicle.
If caught without auto insurance you can be ticketed or even have your vehicle towed and impounded until you get auto insurance and pay the impound and towing fees which can cost more than the auto insurance itself.
Although there is a state such as New Hampshire that does not require all drivers to have auto insurance but you still are responsible for paying any damages that occur due to your fault.