Your mother asks you to pick up your room. You refuse: you demur. Your friend wants to go to the Death Metal Forever concert, but you hesitate: you demur. Whether you strongly object, politely disagree, or hesitate to agree, you demur.
If Aunt Tilly offers to knit you a sweater, you might politely demur, being reluctant to accept. When she describes the bunnies she plans for the sweater, you would want to strongly demur, explaining that you plan to move to Texas next week and will no longer need sweaters. And if you find yourself the defendant in a civil suit, you might file a demurrer to object to the plaintiff’s complaint. When you file that demurrer, you demur.