When apersonagrees to do something or tonot do something, that person is making apromise. A promise can be madeverbally (by saying it), or it can bewritten down as acontract.Breaking a promise, or not keeping it, is often just badmanners, but it can sometimes beillegal, such as when a contract is not kept.
Avow is a special promise. It is mostly used in areligious sense or inceremonies such asmarriages when the couple who are being married make their "marriage vows", promising to befaithful to one another.
Anoath is a promise in thelegal sense. When someone has to giveevidence in acourt of law they "swear an oath". This means that they promise they will tell thetruth.
