If you were at fault in the accident which took place while there was a lapse in your auto insurance coverage then you would normally be held personally responsible for paying for the damages you have caused to others.
It may be that the other party negotiates with you for the cost of their damages or if they have an insurance policy that will cover the damages to them and their vehicle then their insurance carrier may subrogate, come after you for the monies they pay out on the claim associated with the accident that you are at fault for.
If instead you were not at fault in the accident then in many states you could make a claim with the at fault party's insurance provider, even though you were uninsured at the time of the accident. There are some states though that have "no pay, no play" laws meaning that you cannot make a claim for damages unless over a certain monetary amount or for non-economical items such as pain and suffering.
If you were cited for driving without insurance then getting your car insurance reinstated may help your case when you go to court however you likely to face other penalties such as a fine and depending upon state laws possibly your car's registration being suspended for a certain amount of time.