A promissory note is a legal document in which a person or institution promises to pay a debt. You could call a promissory note an official I.O.U.
The term promissory note is often shortened to note, and it's used by attorneys and financial experts like accountants and mortgage brokers. Paper money actually functions as a promissory note — originally, that's exactly what a bill was, a note issued by a bank promising to pay, in gold or silver, the person holding it. The word promissory means "implying a promise," from the same Latin root as promise.