If your mother asks you to clean your room and you refuse, you demur. And if your friend invites you to the Death Metal Forever concert but you hesitate, you demur. Whether you object, politely disagree, or hesitate, 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 are moving 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, objecting to the plaintiff’s complaint. When you file that demurrer, you also demur.