When a person agrees to do something or to not do something, that person is making a promise. A promise can be made verbally (by saying it), or it can be written down as a contract. Breaking a promise, or not keeping it, is often just bad manners, but it can sometimes be illegal, such as when a contract is not kept.
A vow is a special promise. It is mostly used in a religious sense or in ceremonies such as marriages when the couple who are being married make their "marriage vows", promising to be faithful to one another.