Air[]
Air increases your chance to dodge incoming attacks or traps as well as increases your chance to run away from difficult battles. Air also contributes half of its level to your crit bonus.
Earth[]
Earth increases your health by adding its level to your health, then adding a percentage based on its level.
Fire[]
Fire increases your maximum damage by its level, then adds a percentage based on its level. It also contributes half its level to your crit bonus.
Water[]
Water increases the amount of healing that potions and food do.
Luck[]
Luck, though not typically thought of as an element, is in the same category. It has a more widely varied application and adds its level to a lot of calculations such as chance to run away, dodge, the amount of gold you loot as well as your chance to solve a puzzle box.