i used to sell cars, and my gradnparents *always* lease.
what kind of car do you have? how many miles are on it? when exactly did you get it (month and year)? do you remmebr the term? (most leases are about 36 months. dh's lease is 48 months.) why exactly do you no longer want the car?
generally, in the lease agreement, you agree to make all payments and turn it in or buy it out by whatever date is in the contract. let's assume that its a 36 month lease and that you've had the car for 16 months and that your payment is $250 a month. if you turned the car in today, then you're turning it in before the end of the lease term. however, you are still required to make the additional 20 payments at $250 each, totaling $5,000.
i worked for saturn, and when i was there, we had the "lease pull ahead program." what it means is that if you had any GM vehicle (including saturn, chevy, buick, cadillac, oldsmobile, pontiac, and maybe the program included saab and hummer) you would be able to get out of your lease early without any penalites. however, you had to either purchase or lease a new GM vehicle. they did it a few times when i was selling cars, and the people who were eligible usually had a lease ending in the next 9 months or less.
call up the leasing company (gmac, ford motor credit, etc) and ask them what the buyout on your lease would be. (that's basically how much you would need to pay to own the car. its the difference between the selling price and the total of the payments youve made.) check on kbb.com (kelly blue book) or nada.com (i forgot what it stands for, but its something like kbb) for what your car would be worth on a trade (put it in as good or fair condition, since no dealer will ever give you "excellent"). also find out what the private party value is (in case you want to sell it outright).
if the trade in value isnt too much less than what the buyout is, then i'd start car shopping. use the lease as your trade. what you would be doing is buying out the lease, then immediately turning around and trading it towards something else. this may or may not work since a car depreciates the most in the couple years. but at least that would give you an idea of when you *could* do something.
talk to a rep with the leasing company too. see if its possible for someone else to take over your payments. the rep may have some other ideas. (the number should be in your payment coupon book.)