Life is a statistic that measures your monster's health, and determines how many hit points you have. Hit points can be seen in the bar at the top of the screen while a monster is fighting a battle. The more life or HP the monster has, the longer it can last in battle without being K.O.'d (knocked out). If a battle ends before either opponent is unconscious, the percentage of who lost more is figured up to decide a winner, so the more life you have the more chance there is that you'll win.
Note that Life and Lifespan are independent from another. Increasing the Life statistic of any monster doesn't mean that its lifespan will also increase. It's easier if you imagine "life" as "stamina", and "lifespan" as "longevity of life."