Elemental damage is a modifier that exists on certain attacks and weapons. Elemental damage is one of the manifestations of Light, and can be used to a player's advantage to make their attacks more effective. It comes into play for both Guardians and enemies.
There are four types of damage types:
Weapons can be infused with elemental damage; by default, most weapons only do kinetic damage, but some may come with modifiers. The modifier is indicated not only on the weapon's stats but also in the weapon menu on the HUD. At the common level, only Fusion Rifles do elemental damage, but at higher levels nearly all special and heavy weapons do elemental damage. Only a handful of primary weapons do elemental damage, and most of those weapons are typically acquired in Raids.
‡ Two-Tailed Fox fires a Solar Rocket and a Void Rocket, but it’s in-game element is void.
Subclass Elements in Destiny
Subclass Elements in Destiny 2
Elemental damage will do bonus damage against certain types of enemies. The most obvious way to tell if an enemy is weak to a certain damage type is to see what color shield they have; for example, Fallen Captains with blue Absorption shields are weak to Arc attacks. Every race has at least one type of enemy shielded by default but will also have another type of enemy that's shielded whenever the Epic modifier is active (or during certain public events.) 
In addition to the below list, Nightmare Hunts will also frequently spawn large groups of enemies (typically the lowest class associated with the Nightmare's original race, which normally should not be able to be shielded) with Solar shields that are unusually resistant to even matching elemental damage. Nightmare Breaker mods, obtained from the gamemode, are required to effectively penetrate these shields.
† Indicates enemies that alternate between different shield types.
Enemy weapons and abilities typically have their own elemental damage types as well. Normally this doesn't come into play unless players are playing a mission with damage modifiers on. For example, a Nightfall mission with a 300% boost to Arc damage applies to both Guardians and enemies, so enemies that do Arc damage are especially dangerous. Note that Taken versions of enemies will do a different type of elemental damage than the original base enemy.
List of appearances