Contact your insurance company
If you and the other driver decide not to pay for your own repairs, contact your insurance company immediately. If you don't report the accident to your insurance company, and the other driver does report it to his or hers, it could work against you if the case ends up in arbitration.
The claims department of your insurance company will ask you some detailed questions about the accident, and you'll have to provide the information you received from the other driver. They'll also want to know if the police were called, and, if so, the number of the police report.
Your insurance company will then assign your case to an adjuster, who'll be responsible for researching the claim. The adjuster may contact you with further questions, and will probably want to either see the car or a photo of the car, or will ask you to bring the car to an authorized repair shop.
If the adjuster determines the fender bender was your fault, it usually means your insurance company is responsible and must pay for damages to both cars. If the other driver is found to be at fault, usually his or her insurance company pays for damages. If no fault is determined, or if both drivers share the blame, then each insurance company potentially pays half.
If you have insurance, and the other driver doesn't, and it was his or her fault, your insurance company may cover your repairs. Or, you may have to try and recover expenses from the other driver directly. You can start by sending him or her a letter, and a copy of the repair bills or estimates. It may be necessary to work out a payment schedule with that person. If he or she refuses to pay, your only recourse may be to file a civil lawsuit.
Note: Take color photos of the damage done to your car. This can serve as valuable evidence.