From Diablo Wiki
"Elemental" is a term used to refer to non-physical types of damage in games like Diablo 3. In Diablo 3 this term is largely generic, since the different types of damage have no special effect other than graphical. The only exception is cold damage, which can chill and slow, or freeze, targets.
Elemental damage types in Diablo 3 include fire, lightning, poison, cold, and arcane. Other damage types such as Acid and Holy may be classified as "elemental" but that is something of an academic debate over the terminology. These elements can be found on items, with damage types set by the weapon or modified by other gear, or by the type of damage dealt by a character skill.
Though elemental effects in Diablo 3 do not have a major impact on play, they are very noticeable in the graphics with poison leaving behind smoking green corpses, fire blackening enemies, lightning leaving behind charred crisps, and so on. Monster deaths caused by critical hits often trigger special explosions depending on the type of elemental damage, and can be visually spectacular.
Elemental Damage and Secondary Effects
During Diablo 3's development the various types of elemental damage had different special effects from their base damage and/or their critical hits. These secondary effects were removed during development as part of an overall simplification of the game, save for cold damage which retained a chilling or freezing effect, which came at the cost of slightly lower damage.
The Reaper of Souls expansion is likely to reintroduce secondary elemental damage effects on critical hits, though this may change during further development.
A related issue is monster resistances. Unlike earlier games in the Diablo series, monsters in Diablo 3 do not have varying resistances to different types of elemental damage. No monsters are more or less susceptible to fire or lightning or arcane damage, etc. This may change in Reaper of Souls to give different types of elemental damage more impact on gameplay.
Monster Elemental Damage
As with elemental damage from characters, monster damage is basically just divided up into different colors of damage, though these at least impact upon different character resistances. Though some monsters do have different elemental attacks, the main elemental impact comes from Elite Affixes such as Molten or Desecrator (fire), Frozen (cold), Plagued (poison) and others. The most dangerous elemental type from an Elite Affix is generally considered to be Arcane, since the rotating needles can be quite hard to avoid during combat.