The verb demur means to object, to hesitate, or to voice opposition. As a noun, demur means an objection, hesitation, or delay.

The adjective demure means modest, reserved, or shy--or seemingly modest or shy.

Examples

• "If you do have to rebook your outbound flight, make the agent reconfirm your return flight. . . . If the agent demurs, insist and take down all pertinent information—confirmation number, name of agent, time of discussion etc."
  (Catharine M. Hamm, "Thanksgiving Travel." Los Angeles Times, November 25, 2013)

• "She'd invited him for the day's festivities, beginning with the play and then a private luncheon, to be followed by the rest of the San's holiday program—marching bands, picnic supper, fireworks and he didn't know what else—and the invitation had been anything but casual. She'd insisted. Demanded. Required his presence. And he didn't dare demur, because she'd offered an inducement far sweeter than mere duty: money. Hard cash."
  (T.C. Boyle, The Road to Wellville. Viking, 1993)
• "This was not fey and demure Nigella the gastro-porn siren from the television. This was up-for-a-fight Nigella and right from the first bell she was in the mood to slug it out."
  (Paul Cheston, "Sketch: Nigella, the Steely Queen of the Kitchen, Dishes Dirt on ‘Mr Saatchi’ as She Settles Old Scores." The Independent [UK], December 4, 2013)
• "Men have loved to dwell, in fiction at least, on the demure maiden whose inevitable reply is: Oh yes if you please, kind sir! The demure maiden, the demure spouse, the demure mother—this is still the ideal. A few maidens, mistresses and mothers aredemure. A few pretend to be. But the vast majority are not."
  (D.H. Lawrence, "Cocksure Women and Hen-Sure Men." Late Essays and Articles, ed. by James T. Boulton. Cambridge University Press, 2004)

Usage Notes

• "To demur is to hesitate or object because one has doubts. Demure is an adjective meaning modest or dainty. Demurrhymes with fur, demure with pure."
  (Margery Fee and Janice McAlpine, Guide to Canadian English Usage, 2nd ed. Oxford University Press, 2007)
• "DEMURorDEMURE
  A verb and an adjective which look alike. They are not connected, although both have a suggestion of mildness about them.
  
  "A fairly unusual word, demur--pronounced to rhyme with 'purr'--means to 'disagree with.' It does not suggest a violent objection, and is often found with 'not' as if to indicate that the speaker didn't feel strongly enough to disagree . . ..
  
  "Demure--rhyming with 'pure'--is an adjective which means 'modest,' with the suggestion of primness . . .."
  (Philip Gooden, Who's Whose: A No-Nonsense Guide to Easily Confused Words. Bloomsbury, 2004)
• Pronunciation
  "demur di-MUR. Do not say di-MYOOR.
  
  "Demure (di-MUR) and demure (di-MYOOR or -MYUUR) are frequently confused. The verb to demur means to object, take exception to, as in 'He accepts and you demur.' The adjective demure means shy, reserved, or affectedly modest, coy. Many educated speakers slip up by saying demure when they mean demur. This is an error of overrefinement, like saying AF-tur-NYOON for afternoon. Take care not to insert a y sound between the m and u in demur: di-MUR. In demure, however, there is a y sound: di-MYOOR or di-MYUUR. Always stress both demur and demure on the second syllable."
  (Charles Harrington Elster, The Big Book of Beastly Mispronunciations: The Complete Opinionated Guide for the Careful Speaker, 2nd ed. Houghton Mifflin, 2005)

Practice

(a) "It was his idea that he move into one of the guest rooms that night. I didn't _____--it was obviously the only thing to do short of one of us going to a hotel."
(David Ambrose, Coincidence. Macmillan, 2001)

(b) "Her face displayed, along with that _____ quick smile he could now remember--a smile that darted in and out--a good sense of herself, an established social identity momentarily set aside, for this occasion, like a man's jacket folded into an airplane's overhead bin."
(John Updike, "The Walk With Elizanne." My Father's Tears and Other Stories. Alfred A. Knopf, 2009)
 
Answers to Practice Exercises

Glossary of Usage: Index of Commonly Confused Words

200 Homonyms, Homophones, and Homographs