A two-handed weapon (or abbreviated as 2H-sword/mace/axe) is a weapon that requires both the main hand and off-hand slots in use (as opposed to one-handed using one slot). Two-handed weaponry includes swords, maces, axes, staves, and polearms. Generally two-handed weapons are slower than their one-handed counterparts, but balance it out with stronger stats.
If the character equips a two-handed weapon, any item in the off-hand slot will un-equip. The off-hand is not usable when a two-handed weapon is equipped, even though the slot will appear empty. You can equip a two-handed weapon and a ranged weapon at the same time, as they use different slots. Warriors are the exception, however; due to Patch 3.0.2 they can dual wield two-handed swords, maces, and axes (spears and staffs are excluded) when they obtain Titans Grip, a Fury ability. Rogues can not use two-handed weapons of any kind.
Mages, priests and warlocks can use two-handed weapons in the form of staves. Paladins cannot use guns or bows, thus rendering this point moot for them too.
Eligible classes no longer need to see a weapon trainer to learn this skill.