Monsters in the Battle Arena can have one of 8 possible elements, or none at all, as seen in the Bestiary. The elements are in a chain: Fire is strong against Ice, Ice is strong against Wind, Wind against Earth, Earth against Thunder, Thunder against Water, and Water against Fire. Light and Dark are outliers, in that they are strong against each other. For each 1% you have in Fire element, you will gain 1% additional Attack when fighting Ice-element monsters, and 1% additional Defense when fighting Fire-element monsters.
For the most part, the only elements that are used are Light and Dark elements. This is because in the Dark Pyramid and Spooky Crypt zones, the monsters are all Dark-element. To maximize damage against these monsters, you should stack Light Element to increase your attack, either through skills or Green (A) Bonuses. To help survive the damage from these monsters, you should stack Dark Element to increase your defense. This bonus attack and defense, unlike straight Attack % Bonuses, will NOT show up on your stat screen. This means that your actual damage dealt will be higher than shown.
Some monsters have no element listed. In this case, using Element Bonuses will have no effect either way on these monsters.
It should be noted that Green Bonuses are all reward based with the exception of Element Bonuses. This can mean that you are sacrificing rewards to increase damage.
Never use the All Elements bonus.