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As noted above, a lease is a long term rental contract. You pay monthly "rent" on the difference between selling price (equivalent to the negotiated purchase price) and the residual value at the end of the lease term (typically a percentage of MSRP) plus interest. Negotiating points are generally the sale price (make sure to include in the agreement that all cash incentives further reduce the sale price - in other words, are for your benefit) and money factor (aka interest rate); the residual value (% of MSRP) is probably non-negotiable if you're leasing from a manufacturer or its captive financing arm.
Understanding the numbers, and looking out for dealer profit centers like miscellaneous fees and interest rate mark ups, will result in a better deal than trying to negotiate a particular monthly payment. There are numerous lease calculators online, make sure to double check the dealer's monthly payment amount; you will need to find out how your state handles sales tax on leases (some tax the entire value of the car up front, others apply tax to each payment, etc). Also, for leases you are generally better off not putting money down as a "capital cost reduction" - if the leased car is totaled while driving home from the dealer, that money is typically lost (gap insurance generally only covers the leasing company); no money down does mean a higher monthly payment, but you can invest the "down payment" money and use that to offset the increased payment. Lastly, note that some manufacturers give a discount on the interest rate for providing additional security deposits.
A few years ago the Mrs. asked for egged pizza for her birthday dinner. Pie came off the egg looking great, until I tried to remove my "crutch". That was when I learned the difference between parchment and wax paper, which are NOT interchangeable for this purpose . . .