1. My three-year-old son has become rude and demanding. I don’t like his bad behavior and I’m going to nip it in the bud before it gets any worse.
2. The boss wasn’t happy with a few of his workers who had begun to leave the office before quitting time. He was afraid they would fall into the habit of leaving early if he didn’t nip it in the bud, so he told them they would have to stop.

The expression compares stopping a bad situation to cutting (nipping) a flower before it has a chance to grow (while it is still a bud).