The rules, procedures and laws regarding vehicle losses vary from state to state and from insurance company to insurance company. This situation is an individual experience and every situation is unique, but it is good to understand your rights.
In most states, insurance companies have the authority to declare a vehicle a "total loss". If they decide to allow you to keep the vehicle, the word "salvaged' may appear on your title in which case it would be illegal to operate the vehicle on the roads and highways.
A car is "totaled" when it is so damaged that it would cost more to fix it than what it's actually worth. A vehicle is only worth the Fair Market Value determined by the insurance company using procedures defined and set forth in your policy. If you don't agree on the ACV or FMV your insurer determines, you can bring in an independent appraiser at your own expense or seek the advice of an attorney.
Some companies consider a damaged vehicle a total loss when the total cost to repair it exceeds just 51 percent of the vehicle's actual cash value (because of added storage costs, rental car, etc.), while others don't consider it a total loss until it is at 80% of the ACV.
It can be hard to decide if you want to take the money they are offering or keep the car and have it repaired. If you decide to keep the car and have it repaired (if that option is available), typically you can take the ACV, minus deductibles and what the insurance company would have received at the salvage yard, and fix the car yourself.
Comment Update: We have added more text to help explain your rights.
Comment Update: Re: Difference between it being your own insurance company, with whom you have a contract, and someone else's.
You are not obliged to accept their terms, but typically that would preclude them from settling with you. The other company has a responsibility to make you whole and you always have the option to not accept their offer.
Comment Update: Please read our related articles on GAP Insurance.