Fatigue is a (partially hidden) factor in your monster's health in the video games, and runs on a 0 - 100 point scale. It is essentially the depletion of your monster's energy, and it is the physical equivalent to Stress.
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, pass out, or in some games, die in battle.
If you monster is injured due to fatigue, you may need to take it to the Hospital.