You usually have someone notarize your contract when you lease a car. In other words, she'll put a special, official stamp next to your signature and her own — certifying that you are in fact you.
The verb notarize is most likely to come up in a lawyer's office or when you're signing a contract, a will, or some other legal document. A person certified to notarize documents is called a "notary public," and his job is to officially swear that he saw the contract or agreement get signed. The word "notary" was originally used to mean "secretary," but around the 14th century it took on the meaning of "person authorized to authenticate."