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).
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. 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. In Diablo III, many items can roll increased damage of all skills of a specific damage type. In Diablo I and Diablo II, they were collectively named Magic Damage. Note that Arcane and Holy count as one damage type for the purpose of Resistance. Elemental Damage is a collective term for all seven (eight) major types of damage used in the Diablo series.