They cut you a check, they take the car. Don't know if it's different outside the states but I'd assume it's all the same.
You might have to negotiate a bit on the value of the car, but if you feel you're getting a fair price that's that. You get a check to pay off what's owed (if any) and the left-over you use to buy a new one. If it's paid off, you're good to go.
They "total" a car based on the value of the car pre-accident, and the amount of money it would take to fix. If the damage is more than a certain % of the value of the car (I think it's somewhere between 70-85%) they'll cut their losses and just consider it a total loss.
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