To be displeased or offended by the actions of others.
There doesn't seem to be much we can do with umbrage other than to take it, i.e. become displeased - the word is no longer used in any other context. What is umbrage exactly? It sounds like some form of distasteful patent medicine.
Step back to the 15th century and umbrage didn't mean displeasure. The word was inherited into English from the Latin 'umbra', meaning shade. Umbrage came to be used in English to mean shade or shadow, or the foliage of trees which cause shadows; for example, this piece from John Lydgate's 1426 translation of De Guileville's Pilgrimage of the life of man:
...my vysage whiche is clowded with vmbrage,
'Taking umbrage', i.e. sitting under a shady tree, had then no negative associations, as is made clear in Sir Thomas Elyot's The image of gouernance, 1540:
The sayd trees gaue a commodyous and plesant vmbrage.
Over time, the figurative use of umbrage to mean displeasure evolved, probably from the simple association of darkness with gloomy thoughts. In that meaning, umbrage was first said to be given rather than taken, as this example from Sir Nathaniel Brent's 1620 translation of the Historie of the council of Trent shows:
He... therefore besought them to take away all those words that might give him any Vmbrage.
The shade/disfavour metaphor is made explicit in this piece from Sir Robert Naunton's Fragmenta regalia, 1635:
On the fall of the Duke he stood some yeers in umbrage, and without imployment.
The first record of anyone taking umbrage is in Lord Fountainhall's [Chronological Notes on] The decisions of the Lords of Council and Session, 1680:
The Bishop... took umbrage at his freedom of speech in the pulpit anent [side by side with] the government.
J. K. Rowling picked up on these associations when choosing the name of the unpleasant character Professor Umbridge in the Harry Potter series. The negative link was reinforced with the choice of Dolores (from the Latin dolour - pain) as first name.