Promise
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.
An oath is a promise in the legal sense. When someone has to give evidence in a court of law they "swear an oath". This means that they promise they will tell the truth.
Promise Media
American immigrants seen swearing the Oath of Allegiance in 2010
Linking little fingers is one of several common gestures indicating a promise has been made.