When you were little, heading out the door on a freezing cold day, your mom probably made an exhortation to put on your hat and zip your coat up all the way. An exhortation is a loud or enthusiastic urging.
Use an exhortation any time you really want to encourage someone to do something. It's a way of communicating that's persuasive and powerful — an exhortation might come from a government in the form of a speech encouraging people to get flu shots, or it might be a teacher urging his students to sit quietly. Whether it comes in the form of a warning or encouragement, an exhortation strongly advises people to take some particular action.