The Lifespan of a monster determines how long it will live. It is usually measured in terms of weeks. The lower a monster's lifespan, the shorter it will live. Factors that affect a monster's lifespan are stress, fatigue, and the nature of its main breed. The amount of battles it sees or how many explorations (or depending on the game, the number of errantries) it goes on can also shorten a monster's life. It also depends on what you feed it and how well you treat it.
In some games, monsters don't die at all. They simply retire at a certain age and are sent to work as coaches for the Monster Associations.