If my wife was rear-ended and the other party has no insurance, what will happen to the other party if I file through my insurance?

If your wife has uninsured motorist coverage on her auto insurance policy, she can file an uninsured motorist claim with her insurance company.  If she does not have uninsured motorist coverage, she will need to file a lawsuit for negligence against the uninsured driver and registered owner of the vehicle (if different than the driver).  Your wife's damages (the amount she is seeking to recover in her lawsuit) will be the cost of repairs to her car.  She will need to mitigate (minimize) damages by selecting a repair shop whose charges are comparable to other auto repair shops in the area.  If she selects the most expensive repair shop she can find, her damages will be reduced accordingly.  If a rental car is needed while her car is in the repair shop, again it will be necessary to mitigate damages by not selecting the most expensive rental, but selecting a rental car with reasonable rates.  In addition to the cost of repairs and a rental car, damages should include court costs such as the court filing fee and process server fee.  A lawsuit can be filed in the state where the plaintiff resides or in the state  where the defendant resides or in the state where the claim arose.  For convenience purposes, such as filing documents with the court and court appearances, the case should be filed where you and your wife reside.  Since you said the damage was minor, it may be possible to file the lawsuit in Small Claims court.

If your auto insurance policy includes uninsured motorist coverage, your auto insurance company will pay for the repairs to the car.  The auto insurance company will sue the uninsured driver to recover the cost of repairs espended plus attorney's fees.