If you promise that you will do something, you say to someone that you will definitely do it.

If you promise someone something, you tell them that you will definitely give it to them or make sure that they have it.

A promise is a statement which you make to a person in which you say that you will definitely do something or give them something.

If a situation or event promises to have a particular quality or to be a particular thing, it shows signs that it will have that quality or be that thing.

If someone or something shows promise, they seem likely to be very good or successful.