Classes other than Wizards, for the most part, do not deal Arcane damage: their only source of it would be equipping weapons with bonus Arcane Damage, therefore they will not deal pure Arcane damage, unless they are insanely bold or suicidal enough to simply use a basic attack. Some special effects, like Moonlight Ward, also deal Arcane damage. Eirena's spells use Arcane damage type as well.
All Wizard spells that do not rely on elements will use Arcane damage type, even the Diamond Skin. Spells associated with time (such as Slow Time or Temporal Flux), force (Explosive Blast or Wave of Force) displacement (Teleport or Mirror Image) and cosmos (Archon or Black Hole) will likewise use the Arcane damage type.
Andariel's Visage, Moonlight Ward, Stone of Jordan, Lidless Wall, Swamp Land Waders and Gesture of Orpheus are the items with unconventional bonus to Arcane skills damage. Countess Julia's Cameo amulet will render the character completely immune to both Holy and Arcane damage (the attacks will actually heal the Nephalem).
The Arcane damage type was revealed by the Witch Doctor in the Diablo III gameplay video at WWI08. The Witch Doctor was seen wielding a sword imbued with Arcane Damage because of its characteristic purple glow.
After the revealing of the Wizard at Blizzcon, one of whose skill trees includes Arcane Skills, mechanics of Arcane damage were slightly elaborated. Upon inflicting a critical hit, Arcane Spells were to silence opponents, and (with the right synergies) slow them as well. Possibilities of extra damage to spell-casters or doing damage to the opponent's mana pool were discussed as well, much like the Arcane Towers seen in Warcraft III: The Frozen Throne.