The off hand, or off-hand, or offhand, is the second weapon slot all characters have access to in Diablo 3. It's where the "left weapon" goes when a character is dual-wielding.
The Barbarian, Demon Hunter and Monk can dual-wield weapons, and the Monk is limited to doing so with two fist weapons, all characters can hold some other objects in this slot.
Shields are held in the off hand. The Wizard can hold an orb in that hand, teaming it with a wand or a dagger and the Witch Doctor can hold a Mojo.
None of the characters in Diablo 3 can use two-handed weapons with one hand, so whenever anyone is equipped with one, they do not have the option of an off-hand weapon.