When your car is totaled, the insurance company has an obligation to "make you whole," as that is defined in the policy. This essentially means you have to be left in approximately the same financial position you were in before the accident. To accomplish this, the insurance company will typically write you (or you and your lien holder if the vehicle is financed) a check for the actual cash value (ACV) of the vehicle, minus any deductible on your policy. After the settlement is paid, the damaged car goes to a salvage yard, where it is typically auctioned to the highest bidder and used for parts and this is amount they term the salvaged value of the car. The insurance company keeps the proceeds of this sale to recoup some of the monies paid out on the claim.
If you want to keep your damaged vehicle, some insurance companies will forgo the auction process and turn the car over to you (usually in cases where the car is over 10 years old). They will still have to pay you the actual cash value of the car, minus any amount the car would have brought at auction (the salvage value). At that point, it is up to you to pay for the necessary repairs. If your insurer allows you to do this, you will have to inform your insurer right away if you want your car back. Once it goes to the salvage yard, you'll have little chance of getting it back, since only licensed auto salvagers are normally allowed to attend these auctions.
Even if your insurer allows you to keep the car, it may not be worth the time and expense to get it back on the road if your state has a number of special requirements you must satisfy. Many states require you first obtain a salvage title then after repairs are made that you have the car inspected by the state to show it is roadworthy and then obtain a rebuilt title. Also, some insurance companies will not insure vehicle with a branded title meaning salvage or rebuilt title.
Remember, the check you receive from your insurer is for the ACV of the vehicle, which may not match the cost of a similar car in the real world. If you think the settlement amount your insurer offers you is too low, but you don't want to go to the trouble of having the damaged vehicle repaired, you may be able to negotiate a higher settlement. To do this, you'll need to bring in an independent appraiser (probably at your own expense). If this appraisal is significantly more than the insurer's internal appraisal, the insurer may agree to increase your settlement.
If you do not want to pay for an appraiser you may first try seeing if documentation from local dealers about the value of your car being more than the offer will help you negotiate a higher settlement amount for the totaled out vehicle.
Your options include agreeing on a settlement amount, letting the other insurance company keep your vehicle and buying a different one or agreeing on a settlement amount and buying back the vehicle then getting the branded title on it and repairing the vehicle to make it roadworthy. An insurance company once it find its not economical to repair a car and finds it a total loss will not normally reverse its decision and fix the car, instead if you want it you will need to buy it back and repair it on your own if that is possible according to state laws and the insurance company's guidelines.
To find out about your state's laws regarding salvage titles contact your Department of Motor Vehicles.