To anger someone in a superior position severely; to incur the wrath of one's boss or superior — an idiomatic expression.
軽率な発言で上司の逆鱗に触れてしまい、評価が下がった。
My careless remark angered my boss, and my evaluation dropped.
会議で彼の計画を公然と否定したら、社長の逆鱗に触れた。
When I publicly rejected his plan at the meeting, I incurred the president's wrath.
目上の前で失礼な言動をすると逆鱗に触れることがあるから注意するべきだ。
You should be careful with rude behavior in front of superiors since it can provoke their anger.
Closely matching the original meaning: to offend the emperor or an absolute authority, or figuratively to touch the dragon's throat scale (said to bring divine punishment). This sense carries historical/literary nuance.
古い史書には、家臣が逆鱗に触れて罰せられた話が記されている。
Old chronicles record stories of retainers who offended the ruler and were punished.
民衆が王の逆鱗に触れる行為は、古代では命取りになりかねなかった。
In ancient times, actions that offended the king could be deadly for the people.