Mana Globes are orbs of magical energy that function much like Health Globes; they restore mana (25% of the maximum as a default, though some character skills can boost this total) when a character runs through them. As you might expect, they look just like health globes, but are blue, instead of red.
Mana globes were removed from the game prior to the Blizzcon demo build in August 2009. Wizards and Witch Doctors who formerly benefited from mana globes now gain mana (or instability) from health globes, if points are placed in the skills that enable this benefit.
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How they Dropped
When mana globes were a feature, they only dropped when a Witch Doctor or Wizard was in the game. Wizards had a (now removed) skill called Mana Recovery which gave a % chance for a dead monster to drop a mana globe. Mana globes did not benefit Barbarians, since they did not provide any boost to their Fury resource.
Removal Explanation
The Diablo 3 team explained that while mana globes were functional, they weren't necessary. Only Wizards (prior to their mana becoming Instability) and Witch Doctors were able to use mana globes, and this added needless complexity to the game. It was easier to make health globes replenish both health and mana. This it not always the case; Wizards and Witch Doctors must spend some points in the various skills that enable them to gain mana from health globes before this function is enabled.
Giving health orbs that double purpose also made the health orbs more valuable to all characters. Jay Wilson said that prior to the change, Witch Doctors often had no use for health orbs, since they weren't taking much damage while staying behind their tanking pets. Post-mana globes, Witch Doctors need regular health orbs, since even if they aren't low on hit points, they generally want the mana boost.
Who Benefits?
When a health globe is consumed, all characters within range receive the bonus. This includes the bonus to mana (if any) a character would gain if they consumed the health globe themselves.
It's unknown if there will be any benefits to the Barbarian's Rage or the Monk's unnamed non-mana resource. The potential mana-like benefits to the unrevealed fifth character are likewise unknown.