Chastise is a fancy word for telling someone that something they did was really bad. If you pick your nose, your mom will probably yell at you. If you pick your nose in front of the Queen of England, your mom will chastise you.
Back in the Middle Ages, chastise used to also come with a beating — that sense of the word has passed, and in fact, people tend to use chastise when they are trying to accuse someone else of overreacting. "You're chastising me for forgetting to feed the cat, but it's not like the cat died!"