At someone's behest means at their request, command, or urging. When you do something at your teacher's behest, you're doing it because they asked or told you to. When a group of students organizes a bake sale at the behest of their principal, she's the one who suggested or requested they do it.

The word carries a formal tone and often suggests that the person making the request has some authority or influence. You might read that a knight undertook a quest at the behest of the king, or that scientists conducted research at the behest of the government. It's not quite as strong as an order, but it's more serious than a casual suggestion.

You'll mostly encounter behest in writing or formal speech rather than everyday conversation. A friend wouldn't say “I'm here at your behest.” They'd simply say “You asked me to come.” But in a story, you might read that “the rescue mission was launched at the behest of the president,” which captures both the seriousness of the request and the authority of the person making it.