There are four basic elements within the game fire, wind, water, and earth. Each element has it's weakness and strengths. Fire beats earth, earth beats wind, wind beats water, and water beats fire. What does this mean and how do you use them? You equip them to your weapons or hobby armor of course! By equipping an element to a weapon that weapon is now stronger against a monster if the monster is weak to the element you equip. Ex. Wind element bow > Giant slime. If you attack a monster with the SAME element it will not do as much damage and may take a longer time to kill it.Ex. Fire element sword < Clay Boar. Equipping these to hobby armor will have a similar effect with your defense (although you won't be healed if hit by the same element) it will decrease damage taken by a specific element depending on armor. I'm not sure if you take more damage when hit by the dominant element but I'd assume so.
Here's a list of what monsters drop what elements:
Water element: Giant slime and crabs (Left)
Fire element: Flame slime (Up)
Wind element: cockatrice, basilisks, minotaurus and moths (Right)
Earth element: Desert slimes, minotaurus, and crawlers (Down)