There are three big variables on a lease: capitialized cost (or "cap cost"), money factor, and residual value.
Like tomhaney1 said, not enough people realize that negotiating the price you pay for the car is the same whether you buy/finance or lease. Most people who lease start with a payment in mind, and then negotiate based solely on the payment. But that's bad!
Fundamentally, a lease is a car loan on only the amount of money that is expected to be lost as depreciation over the term of the lease (usually 3 years). But at the end of the lease, the bank owns the car, not you.
The cap cost is whatever you set as the purchase price for the new car. If you trade something in or put money down, that's often known as a "cap cost reduction". Anything you read on the forum here about actual sale prices being near invoice or even lower with rebates still applies to leasing. So start by getting the best purchase price you can!!
The money factor is lease-speak for the finance rate, like the APR on a loan. The money factor is always 1/24 of the APR, regardless of the number of months in the lease. So, for example, 5.9% APR comes out to a money factor of .00246.
The residual value is what the car will be worth at the end of the lease. The dealer can keep the car, or you can buy it out for this price at that time. Too many people think that a low residual value is good, because it will be a "good deal" to buy it at the end of the lease. But that's not true - since your lease payment is roughly based on the (cap cost minus residual value) divided by the lease months, then the lower the residual value, the higher the lease payments. You're paying for that "good deal" over the term of the lease.
Of course, dealers and banks make money on leasing, or they wouldn't incur all of the administration, overhead, and liability of owning cars that other people drive around for 3 years. Where they do that is in the residual value. In order to avoid getting "stuck" at the end with a vehicle worth less than they planned, they often set pretty low residual values. For example, Toyota Financial's standard residual is 53% of MSRP after 3 years with 12K miles/year. I think you can argue that any well-maintained Toyota will be worth more than that in 3 years. So, they're making money when you turn the car back in. If the car is worth 58%, they've immediately made 5% without lifting a finger. So that's the "catch" with leasing. Because of the risk involved to the bank, the residual value is often not negotiable, although you can try to find alternate banks/credit unions that might use a more favorable residual number.
To figure out your lease payments, there's a calculator at
Edmunds.