"Not merely a container for arrows and bolts, the quiver often contains mystic properties. Some quivers provide easier access to projectiles and thus quicker rates of fire; others confer on their arrows the crackle of lightning or the sharpness and impact of thrown spears."— Game Guide(src)
Similarly to Arrows and Bolts from Diablo II, the Quivers may be equipped in the off-hand slot in addition to any other ranged weapons (Bows, Crossbows or Hand Crossbows), even if those weapons are two-handed, but not with dual Hand Crossbows. As of patch 2.4, they can be also equipped with one-handed melee weapons.
In Diablo III, arrows are infinite, so quivers do not limit the number of shots in any way. They are also not shown on a character model, but all pants on Demon Hunters do show quivers in some way, sometimes even two at a time.
All Quivers have a guaranteed Attack Speed increase (usually 15 to 20% regardless of level) instead of flat damage the other off-hand items provide. This affix is free, and does not count towards the total 6-affix limit.
They can roll core and all-armor stats, one Socket (count as Armor for that purpose), Area Damage, Critical Hit Chance, bonus damage against Elites, reduced skill cooldowns, increased damage of any skill, reduced resource costs, increased maximum Discipline, Hatred regeneration, Life %, Life per second, Thorns, extra healing, and various 'chance on hit' effects.
There are no set Quivers in game.