Stress is a (partially hidden) factor in your monster's health in the video games, and runs on a 0 - 100 point scale. It is the psychological equivalent to Fatigue.
Stress usually builds up by doing Drills, and can be effected by certain items (for instance, Mint Leaves usually relieve Stress), or in some games, by your monster's inclination to like or not like certain items (i.e., if it hates Wine but I continue to give it some, it will become stressed.) Even a low level of stress can cause your monster's lifespan to shorten slightly, so be careful!
If your monster becomes too stressed, in most of the games it will run away. You can pay a search party to help you find it in most cases, but it will take away valuable time from Training.