Even though you don't technically own it, you can sell a leased car to a dealer in nearly the same manner that you would for a financed car. There are exceptions to this, so check the terms and fine print of your lease to ensure that this option is available and that you will not incur any large fees. It is also possible to sell the vehicle privately, which can often result in a higher sales price, but this option comes with risks of its own.
You don't own the car, but the money and equity you have put in to your leased vehicle are still yours and they can be used to your advantage when selling or trading it in for a newer car. Although most people tend to go back to the dealer where they originally leased the car, this is not a requirement, and generally you are free to sell the vehicle anywhere you want to. The ability to shop around before selling makes it important to have your car appraised by multiple sources if you're not happy with an offer.
More specifically, selling a leased car involves taking it to a dealer of your choice -- one that gave you a fair appraisal -- and allowing the dealer to buy the car from the company you leased it from. The dealer will then pay you for any equity, and -- although this payment won't come instantly -- you do not have to wait for the dealer to resell the car.
Although this method is not as straightforward as selling your leased car to a dealer, it is possible to get a better price this way if you have a trustworthy buyer lined up. Do not sell to strangers using this method. The private buyer will take the place of the dealer, so he will need to arrange payment with your leasing company and, in turn, the company will give you the title to the car. You will then sign over the title, and the buyer can handle the rest.
Whichever method you choose to use when selling your leased car, be aware of all the fine print and legal agreements with both your leasing company and the buyer, if the two are different parties. It is also important to know exactly how much your car is worth now, as well as how much equity you have, before making the final decision.