In most states if you were at fault then the other party would be able to place a claim against your insurance policy for the damage their vehicle sustained even if they were uninsured at the time of the accident.

The claim would be made against your property damage liability coverages. Hopefully the other driver was ticketed for driving without insurance so that he or she will need to take care of that citation and any penalties associated with it. Penalties can vary depending on state laws but can include fines, license suspension and/or registration suspension.

Now there are some states that do not allow a driver to place a claim due to being uninsured at the time of the incident. For example Louisiana's Insurance Commissioner notes this state’s “No Pay, No Play” provision. What this means is that drivers who do not have the Louisiana mandatory $10,000 liability insurance forfeit their right to use the court system to sue for the first $10,000 of certain types of damages.

So it will depend upon your state's laws if person driving without insurance is able to collect for their damages from you, the at-fault party. Check with your insurance agent or state's insurance regulator to see what state laws allow in your situation.