Black weapons or black damage are weapons that don’t have any sort of elemental damage.
The reason these weapons matter is due to a bug in how Diablo 3 calculates their damage in conjunction with other items that add elemental damage.
Black weapons are the best type of weapons to have when if you have other items that grant additional % elemental damage. For instance:
Zunimassa’s Trail – boots with 7% poison damage
Tal Rasha’s Allegiance – (7% elemental damage)
Triumvirate - (5%/5%/5% cold/fire/lightning damage)
The bug exists because the extra % added damage is calculated from the non-elemental damage on your weapon. So if you have a black weapon that has 100 physical (black) damage then Tal Rasha’s would give you 7% of 100 the full 100. But if your weapon was 100 damage (25 physical/75 elemental) then Tal Rasha’s 7% bonus would only calculate on the 25 physical/black damage.
In early 2013 Blizzard announced that they would not be fixing this bug on existing items, and that while they would fix it someday, there were no immediate plans to do so. Black weapons were not changed in v1.07, though the bug was removed from the bonus provided by socketing rubies in weapons.