Elemental Damage is a category of damage in the Dark Souls games.
Dark Souls
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.
Elemental Damage can be cast with numerous sorceries, miracles and pyromancies, and almost always exclusively deals some form of elemental damage.
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 II
In Dark Souls II, a new elemental damage type in the form of Dark has been added, as well as the Hexes school of casting. All forms of elemental damage are tied to how Intelligence and Faith are leveled up, with different amounts of effectiveness depending on the stat spread. Magic is tied to Intelligence, Lightning is tied to Faith, while Fire and Dark are tied to both.
Dark Souls III
Dark Souls III combines the elemental systems of the previous two games. Hexes no longer exist as their own school of casting, and elemental damage is no longer rigidly locked to how Intelligence and Faith are leveled (for instance, Darkmoon Blade grants Magic damage based purely on Faith).