Summary[]
There are four elements: Fire, Ice, Thunder, and Holy. Every attack is either non-elemental or one of these elements. Having an element adds a multiplier to the damage of anywhere from x1/4 to x2, depending on the defender's elemental resistance. In the case of monsters, x1/4 is instead absorbed, healing the monster for x1 damage.
See Also: Barrier Shift
Equipment[]
For equipment, there are basically two types of resistance: x1/2-x2 and x1/4. The main thing that matters is that x1/4 overrides the x1/2-x2 range entirely. In other words, a Red Robe (x1/4 Fire) combined with an Ice Shield (x2 Fire) would still result in x1/4. By combining opposing shields and robes, you can have x1/4 resistance against up to 3 elements without equipment like Dragon Mail.
Note: You cannot stack x1/4 multipliers to get better than x1/4 resistance in an element.
Multipliers are applied in order of weapon, armor, and then accessory. The combined multiplier cannot escape the range of x1/2-x2, so if a weapon had x1/2, an armor had x1/2, and an accessory had x2, the result would be x1 (x1/2 x 1/2 = 1/2, 1/2 x 2 = 1). There aren't any weapons with x1/2 or x2 elemental resistance, however, so this primarily means you can't combine x1/2 and x1/2 to get x1/4. The only way to get to x1/4 is through a single piece of equipment that provides it or natural immunity.
Lists[]
List of non-elemental attacks
List of Fire attacks
List of Ice attacks
List of Thunder attacks
List of Holy attacks
List of monsters that use Fire attack(s)
List of monsters that use Ice attack(s)
List of monsters that use Thunder attack(s)
List of monsters that use Holy attack(s)
List of monsters and equipment that halve Fire damage
List of monsters and equipment that halve Ice damage
List of monsters and equipment that halve Thunder damage
List of monsters and equipment that halve Holy damage
List of monsters and equipment that absorb/quarter Fire damage
List of monsters and equipment that absorb/quarter Ice damage
List of monsters and equipment that absorb/quarter Thunder damage
List of monsters and equipment that absorb/quarter Holy damage
List of monsters and equipment that take double damage from Fire attacks
List of monsters and equipment that take double damage from Ice attacks
List of monsters and equipment that take double damage from Thunder attacks
List of monsters and equipment that take double damage from Holy attacks