If your vehicle is hit by a person that does not stop you have just been involved in a hit and run. If you have eagle eyes and a good memory you can try to remember the make, model and color of the car but most importantly the license plate number. You should call the police as soon as the incident occurs so that law enforcement that arrive to the scene and take a report. Having information about the driver that drove off should help the police track down the party that damaged your automobile.
With a description of the car in hand, along with the all important and identifying license plate number the police should be able to determine the owner of the vehicle that stuck your car causing damages and sticking you or your insurance company with a repair bill. If the owner and driver of the car can be found then you should be able to make a claim through the at-fault party's insurance policy or if you have already placed a claim through your own insurer then your insurance provider can subrogate (go after) the at-fault party.
You may not always see the person that hits your car but if you are lucky and your car was stuck while you were in a parked you may have witnesses to the accident. While those that saw the accident normally cannot tell you who damaged your car many will have thought enough to write down the type of vehicle that hit you and the license plate number. In this type of incident again you should contact the police and request an officer comes out to make a report.
When the police are informed of a hit and run and the license plate of the vehicle causing the damage is known by the victim or witnesses the police should be able to take this information and then determine the owner of the car with that description and license tag number. It then should be told to you or put on the police report so that when you go for your copy at the police station you will know who to contact to get insurance information for the vehicle.
If the police were not called to the site it may be more difficult to determine the owner of a car by their license plate number. Information laws differ in the various states within the US however in some you can contact your local Department of Motor Vehicles and request information on a car owner by giving their license plate number. There may be certain procedures to follow to give the reason you want the information, etc to obtain such information.
In some states there are government agencies set up to help motorist that were damaged by a hit and run driver. These agencies can be a financial responsibility unit, an agency within the DMV, etc but should help those harmed by a hit and run driver and you have the plate number of the offending party.
For example in Florida you contact the Bureau of Financial Responsibility with the crash report and they should research and reply to you within 10 days. In California you can request the insurance information by contacting the DMV's Financial Responsibility Unit and submitting a form.
In New York one must fill out form FS-25 which is a Request for Insurance Information for NY Registrants Involved in an Accident and is available on at the New York State Department of Motor Vehicles office or on the NYS DMV web site.
State laws, especially privacy laws, do differ so you may have to go through a strict process to show you want information on the driver for insurance purposes or to contact them about personally paying for the damages they caused and not for nefarious reasons.
If you were unfortunate enough to get damaged by a hit and run driver but fortunate enough to have at least spotted and written down the offender's license plate number then remember to follow state authorities to determine the owner / driver of the car at the time of the accident. Law enforcement should be called to the scene and your first source to use and try to find out who the person was that hit you. If the police cannot come to the scene or if they were not called for some reason then you can try to get this information by contacting your state's Department of Motor Vehicles.