1) Private sale
This is the most common way of getting rid of your old car. You put up an advert, for example online or in a local paper, and let people come to view and test-drive the car.
Pros: You'll get the best price and remain in complete control of the sale from start to finish. If you advertise the car online you can reach a wide audience and might not even have to entertain any viewings.
Cons: It might take a while to sell your car and you'll potentially have to deal with timewasters. Putting an advert in a magazine or online can cost up to £50, so if you don't sell it you could lose some money. Make sure to complete the paperwork correctly too, or you could end up with problems. Lastly, if the car breaks down, while the buyer has no rights to pursue you over it they may kick up a fuss.
2) Auction
Entering your car into a local or mainstream car auction can be a daunting prospect but it can also be a quick way of selling it. Potential buyers will view your car at the auction and, when it comes up for sale, bid on it until there is no further interest at which point, if it's met the reserve, the car is sold.
Pros: Selling at auction requires practically no effort and even worn out or damaged cars can be put through. You might even get strong money for your car, especially if it's something desirable or interesting.
Cons: You could walk away unhappy if your car sells cheaply - that's the risk with an auction. In a lot of cases you won't have any control over the sale price, as many auctions won't let you set a reserve unless a car meets a narrow window of age or value related criteria. There are auction fees to be paid too, so don't forget to check what they are.
3) Selling to, or through, a dealer
Some dealers will buy your car off you directly. They will assess it and come up with a price that they would be willing to pay. You can also, occasionally, sell your car through them and they will handle the sale for a small fee and a percentage commission. This removes any hassle and will result in more money than simply trading it in.
Pros: Selling to a reputable dealer is usually trouble-free as they will simply pay you for your car and complete the paperwork correctly. As with trading in there's no comeback to you if anything goes wrong with the car. If you've got a desirable car then the dealer may be keen to buy it off you, especially if he's got someone lined up for it.
Cons: You won't necessarily get a good price for it and don't be saddened when you see it a week later advertised on the forecourt for considerably more than you sold it for.
4) Using an online service
You've no doubt seen the adverts on TV for companies that offer to buy your car. Usually all you have to do is enter your details online and they'll quote you a price. If you want to proceed then either someone will come around to inspect your car, or you'll have to take it somewhere.
Pros: If you're just looking to get rid of your car, they'll give you some money for it. It can also be a good way to get rid of a car that's plagued with problems as they'll usually give you more money than you'd get privately and not waste your time.
Cons: In many instances you'll get a fraction of the true value of your vehicle and very rarely will you get what you were initially offered. You may also find you receive a varying level of service from the different companies, which could lead to an unpleasant experience.
5) Part-exchanging or trading in at a dealer
While not strictly selling, when you buy a new car from a dealer you are usually offered the opportunity to either part exchange your car or trade it in on one of the 'scrappage' style schemes. The dealer will take your current car off your hands and deduct their assessed value of it from your new purchase.
Pros: You'll be able to get hold of your new car while simultaneously disposing of your old one, resulting in minimal fuss and the dealer can't chase you if the car breaks, so there's no liability.
Cons: You won't get a great price for your old car and dealers may fiddle the numbers to make it look like you're getting a better deal than you actually are.