To take umbrage is to feel offended or resentful about something, often something that others might consider minor. When someone takes umbrage at a remark, they feel insulted or slighted by it.

The phrase has a slightly formal, almost huffy quality to it. You might hear that a senator took umbrage at being interrupted, or that your aunt took umbrage at not being invited to dinner. There's often a hint that the offense taken might be a bit excessive: the person who takes umbrage tends to bristle and feel wounded, perhaps more than the situation warrants.

Umbrage by itself is an old word for shadow or shade. The connection to offense probably comes from the idea of a shadow passing over someone's mood, darkening their feelings. Today, you'll almost never see umbrage without take attached to it.