I have a 1999 Dodge Neon, and the bank that gave me the loan said that I can't ell the car until the loan is paid off. It doesn't make much sense to me since I'm not paying off the car. I'm paying off the loan. Do all banks do this.

I have mine with Bank of the West.
Your car loan is secured by the car, meaning that if you don't pay the car loan, they repossess the car. If you sell your car and still owe them money, they won't be able to collect the car if you fail to pay. If you want to sell for an amount that is greater than or equal to the outstanding loan amount, ask them how you can arrange to get the lien on the car released and transfer ownership. If you will sell for less than the outstanding loan amount, you'll need to go into pocket to pay off the rest of the car loan.
You can't sell the car until you have the title to the car. You can't have the title until you pay off the loan.

Now if you can find someone that is willing to pay for the car before you can give them the title that you yet don't have because you haven't yet paid of the loan, then go for it. It is generally difficult to find someone that is willing to spend a lot of time trying to get all transactions performed simultaneously.
The car is known as the "equity" or insurance the bank, i presume National Westminster Bank (NatWest), Needs this because if you hae difficulty paying your loan they will Take the cat ans Sell at an auction to raise the money back. Luckily for NatWest Bankers, if you do have difficulty and they sell your car. if it makes more money. i.e you loan is for £1000 and you car is worth £2000. the bank will take £500 for administration charges and etc and you will get the next £500.
That can't be right…of course they want you to pay out the loan first because then they get all that lovely interest money out of your pocket. But I really think they can't stop you from selling the car if you want to - they don't rule your life! What if it was a dud, or you were moving overseas? You'd HAVE to sell it! Just make sure you sell it for more than you have to pay them out! Call and ask for a pay-out figure!
Good luck!