In Diablo III, many items can roll increased damage of all skills of specific damage type. Normally Bracers and Amulets can do so. A Smart Loot drop for a class will only roll the damage types this class can use, for instance, only Fire, Cold, Lightning and Arcane for Wizards. Convention of the Elements Legendary Ring is unique in that it boosts all damage types fit for the current class, cycling between them and augmenting each for 5 seconds before switching to the next one.
Weapons often roll +(X-Y)-(Z-N) Elemental damage, meaning that each attack will add that amount of damage to the base weapon damage, of the specific type (either of the seven, including Physical, although Cold damage ranges are generally slightly less on lower levels, to compensate for the ability to Chill on hit). Interestingly, each hit of that weapon will have a chance (equal to Proc Coefficient of the skill) to count as having double damage type, if those of the elemental damage on weaponry and of the skill itself do not match. This is important for skills like Elemental Exposure or Tal Rasha's Elements set. It becomes especially evident when you see the Critical Hit death animations.
Many skills and attributes require a specific damage type to be used. It is a popular tactic to stack bonus damage of the same type, choosing skills that use the same damage type (commonly called by the selected damage type name, for example, 'Arcane build' or 'Fire build'), thereby effectively boosting all damage done. Even in single damage type builds, some skills are taken with a specific Crowd Control or stat buffing rune, even if that rune has a weaker damage type (as for those skills, their secondary effects are more important than plain damage).As of Patch 2.0.6., the effective damage with the elemental skill bonuses can be seen in the Details tab, Offense section.
There are five amulets in Diablo III that can make characters entirely immune to one non-Physical damage type (it is advised to carry them around, even if they offer no good stats, in case the character encounters a monster who would prove too powerful otherwise):