When you lease a car, you pay for only a portion of the vehicle's cost, the part that you "use up" during the time you're driving it.
You may have the option of not making a down payment, you pay sales tax only on your monthly payments (in most states), and you pay a financial rate, called money factor, that is similar to the interest on a loan. You may also be required to pay fees and possibly a security deposit that you don't pay when you buy a car.
You make your first payment at the time you sign your contract.
When the lease ends, you may either return the vehicle or purchase it for its reduced resale value.