To lease something is to borrow it in exchange for money. If you lease a car from a dealer, for example, you make payments and you get to drive the car, but at the end of the contract you have to give it back.
As a noun, a lease is both the thing you are contracting to rent (the car itself is a lease) and the contract spelling out the terms of use — in other words, a rental agreement. Be sure to check the fine print on your apartment lease to avoid any unpleasant surprises when you move out — like owing money for painting over that hideous wallpaper in the living room.