At the start of each week, your assistant may tell you that your monster is anywhere from "a little tired" to "about to keel over." This signifies how high your monster's fatigue is. Even a low level of fatigue will cause your monster's lifespan to shorten slightly, so be careful!
Training, Drills, Errantries, Expeditions, and Tournaments cause your monster's fatigue to increase. Feeding your monster certain items and letting it rest will cause your monster's fatigue to decrease. In many of the games, entering a battle or going on an expedition when your monster is fatigued can cause it to get hurt or run away.
If your monster runs away because of fatigue, you can often pay a search party to look for it or wait for it to come home.