A promise is an agreement to do or not do something. Also, when you have potential, you show promise.
If your parents say you can go for ice cream and then it doesn't happen, you'll probably say, "But you promised!" People promise small things, like to visit someone, and big things, like marriage: “Til death do us part” is a promise many people make on their wedding day. Another meaning is to make a prediction, as in, "You'll regret this, I promise!” Also, if you show promise playing the violin, maybe someday you'll play in an orchestra.