Promises are an area of law that can sometimes be foggy. Contracts are a form of promises, and typically legally binding. However, some promises are made hastily and without any legal oversight, making them less likely to hold up in court. In general, it’s a good idea to get a written note and have a lawyer look it over before considering a contract legal.
A notary public typically does two things on every document they sign: Add their signature after the person signing the document, and after one of the major paragraphs. There are two types of major paragraphs, a jurat paragraph and an acknowledgment paragraph.