Once the owner has the title, they can then apply for a new registration and license plate. The registration and license plate fees will vary from state to state.
If the owner does not want to keep the car, they can sell it or trade it in. In either case, the owner will need to sign the title over to the new owner.
So, to answer your question, no, you do not get a check when your car is paid off. However, you will receive the title to the vehicle, which will allow you to sell or trade it in if you wish.