If you say something scornfully, you say it in a contemptuous or disgusted way. You might remark scornfully that your brother's breath smells bad. Ewwww.
You might be afraid to try out your high school French at a fancy restaurant in France, for fear that the waiter might laugh scornfully at you. When a teacher is impatient and lacks respect for a student's ideas, he may speak scornfully. The adverb scornfully means "in a way that's full of scorn," and the noun scorn comes from the Old French escarn, "mockery, contempt, or derision."