Please check your automobile coverages. Uninsured/undersinsured motorist coverage would cover this situation. Also in some states, you have to execute a form at the time of insurance purchase declining uninsured/underinsured motorist coverage. If you were never offered the coverage, you may have a claim against the insurance agent for coverage.
Report the matter to your carrier under your "uninsured/underinsured" coverage. Your insurance carrier will subrogate against (sue) the driver at fault for the money they pay out for your damages. If you have to pay a deductible, or have other uninsured expenses (lost/damaged property in the car, time off work to deal with the accident/recover from injuries, payment of a replacement rental), you can sue the other driver in small claims court.
Not having insurance does not mean that he gets off scott free. Yes you will have to bear the initial cost of the damage but you can still take legal action against him/her. Negligence and fault are separate from liability insurance. Sure its easier if they have insurance but they may have to pay out of their own pocket. From a practical standpoint it may not be easy collecting and you wont get an attorney to handle this but your state should have a pro-se clerk that could help you with your papers. Its really a crime that some people have no insurance and its even a bigger sin that some states permit drivers not to have any insurance. Be glad you were not hurt.
If you had collision insurance you can use that to get your car fixed. You will have to pay the deductible. Your insurance company will go after the other driver to get back what it paid to fix your car and also your deductible. If you did not have collision insurance on your own car, you will have to pay out of your own pocket to get it fixed. You can sue the other driver to pay for the repairs, but if they did not have insurance it becomes harder to get money from them.
You did not mention if you were injured and had medical bills or lost wages. If you were hurt you may make a claim against your own uninsured motorist coverage, if you have it. If you do not have uninsured motors coverage, then you can make a claim against the other driver. But, again, if they did not have insurance it might be difficult to collect from them.
Your insurance company must offer you uninsured motorist coverage when you buy your car insurance policy. If you decline to purchase uninsured motorist coverage, your company must have you sign a waiver and keep it on file. If your insurance company failed to make that offer or document your waiver of that coverage, they may have to provide you that coverage anyway.
You should talk to a lawyer in your state with experience in car accident claims and uninsured motorist coverage issues.