If I was rear-ended and the driver is insured but the car was not, how do I get my car fixed?

I assume that the insured driver is not the registered owner of the vehicle and that is why the vehicle is uninsured. 

If that is the situation, if you have uninsured motorist coverage on your auto insurance policy, you could file an uninsured motorist claim with your auto insurance company to pay for the repairs to your car.  If you don't have uninsured motorist coverage, you will need to sue the regiatered owner of the vehicle that hit you for negligence.  You should also name the driver in your lawsuit as another defendant.  Your damages (the amount of compensation you are seeking in your lawsuit) would be the cost of repairs to your car.  You will need to mitigate (minimize) damages by selecting an auto body shop whose charges are comparable to other auto body shops in the area.  If you were to select the most expensive auto body shop you could find, you would have failed to mitigate damages and your damages would be reduced accordingly.  The same applies to the cost of a rental car (if you need a rental car) while your car is being repaired.  You will need to select a rental car with a reasonable rate to mitigate damages.  If you were to select the most expensive rental car you could find, you would have failed to mitigate damages and your damages would be reduced accordingly.