There are three types of Elemental Damage: Magic, Fire and Lightning. Each have a corresponding defense stat that they are measured against. Because of this, certain elemental damage may be ineffective against certain enemies, while other forms of damage will be very effective.
Weapons that inflict elemental damage almost always inflict a form of physical damage as well. The exceptions are the Moonlight Greatsword, Moonlight Butterfly Horn, Crystal Ring Shield and Moonlight Arrows, which all deal only Magic Damage, and the Pyromancy Flame, which deals only Fire Damage when used as a striking weapon.
Dark Souls IIEdit
Being a similar game in a similar place and era, much of what can be said about Dark Souls also applies here, with the only exceptions being weapons that do not exist in Drangleic. As well as this, the Dark damage type has been added, and is primarily dealt by Hexes and weapons associated with being exposed to evil, such as Sacred Chime Hammer.