Health Globes (AKA Orbs) are red, floating globes of life energy that "drop" from slain monsters or spawn in certain locations (including in the Arena). Health globes are "used" when a character passes through or near one, and grant the same full bonus of healing (hit points) to all friendly characters and minions in the immediate vicinity. Each orb refills a character's health globe for a set percentage of their maximum hit points, with the same bonus granted whether a character uses the orb themselves or shares the benefit from a friendly player activating it.
Health globes are the main source of life replenishment in Diablo III. There are some hit point refilling bonuses on equipment (such as LAEK), but nothing like the omnipresent and overpowered life leech of Diablo II. There are also healing potions which grant instant life replenishment, but they are only for use in emergencies, since they have a considerable cool down time between uses.
During the early development of Diablo III, there were blue Mana Globes as well as red Health Globes, but these were removed in mid-2009, and the classes were given skills that could enable them to gain resource from health globes. Currently, the Witch Doctor, Wizard and Demon Hunter each have a passive skill enabling that.
Health gain varies with the type of health globe, but it is always some percentage (25%, 33%, 40%, etc) of a character's maximum hit points. The hit points are not added instantly, but fill up gradually, over a few seconds. Health globes are thus not an instant cure for death, so an embattled, near-dead character should drink a potion for the instant life boost, since they could still die to an attack taken right after using a health orb.
Health globes only add hit points to a character's maximum life and provide no benefit or bonus above that level. A health globe has no effect when used by a character with full health, though Witch Doctors can receive a boost to their mana from a health globe, if they have enabled one of the traits that grants this bonus. A character with full health can still grant the full bonus to other characters though, so if a companion is low on health and stuck in combat, it's wise to use a health orb to assist them.
The healing effect of a health orb is shared by all friendly players and pets in the immediate area (a radius of perhaps 10 yards; characters who are at the far corners of the screen may not be close enough to benefit). Health orbs are community drops, unlike item drops in Diablo III; all players in the game see all health orbs when they appear, and they can be used by anyone.
Each health orb is consumed completely when used, vanishing from the world. Health orbs cannot be moved; rather, they simply float where they spawned. If no player uses one, however, they will remain floating there for some time, and can be returned to for a later health boost.
Health orbs come in various quality levels. The ones dropped by normal monsters are lower quality and will refill just 25% of a character's hit points. Other orbs will refill more; the ones that spawned in the Arena demo at Blizzcon 2010 were worth 40% of a character or minion's hit points, and Blizzard has hinted that there may even be 100% health orbs out there, dropped only by the most powerful monsters.
Gameplay and Challenge
Health orbs are a major part of Diablo III's gameplay design. The D3 Team has worked to remove the easy ways of healing and escaping from trouble that were so common in Diablo II: no more can characters count life leech equipment, or belts full of potions that can be used instantly and repeatedly, or Town Portals for quick town returns. These changes allow the developers to make the game more consistently challenging unlike Diablo II, where the only way things were hard was if a monster could kill a character in an instant, and that wasn't hard, it was cheesy.
In Diablo III the goal is for a more consistently challenging play experience, and for players to have to stick it out in battles to earn a health globe, rather than just popping back to town. Jay Wilson spoke about this design philosophy in an interview from Blizzcon 2009. 
...We did put health potions back in, but they play a very different role. You can't spam them like you used to, you can only use them about once a map. The purpose of those is to take the edge off the loss of health. "I don't have health, or I've got half health, do I want to use a health potion, or do I want to risk it? Ooh, I've got 10% health, it's not even a decision." That's a really interesting decision, and it makes potion use a fun part of the game.
Boss monsters drop health orbs during the battle, generally at set percentages of their hit points. For instance, once you knocked a monster down to 75% of its maximum life, it would drop a globe or globes. Jay Wilson commented on this in 2009,  "[bosses] would drop health globes at percentages of their health. Rares in particular are almost guaranteed to drop about every 25%."
The Systems: Health page on the official Diablo III site used to include some hypothetical examples of how health globes might have to be utilized in a boss battle.
Major boss fights make unique use of health globes. Each boss battle includes a custom-designed means of utilizing these globes to regain health. For example, in one fight, you might have to split your attention between weakening a dangerous boss and slaying its irritating but ultimately less-dangerous minions in order to get enough health globes to stay standing. In another fight, the boss itself might drop health globes when it takes damage, or you might have to hunt for hidden caches of globes in the midst of battle.
Arena Health Orbs
Health orbs appear in the Arena every so often, on pre-set points. There were four such points in the arena map seen at Blizzcon 2010, all located around the center of the arena. Orbs appeared there, always on all 4 spots, 30 seconds after the start of each round. (Assuming the round wasn't already over.)
Health orbs grabbed in the arena grant their full benefit (40% of max hit points) to all nearby members of the team, as well as minions. This dynamic can be seen in the three-shot series to the right, where the blue team Witch Doctor gets an orb that grants a health bonus to his Mongrels, as well as the blue team Barbarian.
Characters and minions need to be nearby to share in this health bonus; a character off the visible screen will not partake of the healing bonus.
Health Orbs were a strategic factor in the Arena matches; getting them was obviously a benefit, but it was also helpful to grab them even if you weren't low on health, simply to deny the heal to the enemy team.
That the health orb spawn points were right in the middle of the arena (at least on the one map available at Blizzcon 2010) forced the action into the center of the level. Many rounds ended with the last survivor of one team dodging attacks around the perimeter of the level, and while this could extend a round for some time, it was very seldom a way to win, since the pursuers tended to be in the middle and would therefore score some health orbs, while the character running for it had no chance to heal up as they hid.
Blizzard commented on this via @Diablo in April 2011.
They’re on timers right now, which actually makes for interesting gameplay. If they were random you’d probably hover there. With set timers you can go off and game people a little more knowing that they won’t spawn again for X time. --Diablo
The appearance of health globes has improved during development. They were initially just red orbs, like a huge drop of blood floating in the air. Over time this look was stylized, and health orbs now wear a golden frame, almost like a jewel setting or a crown on the top and bottom of the orb, a change that gives them a more oval shape.
Health Globes and Mana
Mana Globes were removed in 2009, with some skills changed to give the Wizard and Witch Doctor mana from using health orbs. Currently, the Wizard, Demon Hunter and Witch Doctor can all receive resource from the globes (through the Power Hungry, Vengeance and Gruesome Feast passive skills, respectively).
If the developers have sketched out some lore for health orbs, they have not made it public. An in-game explanation for health orbs isn't essential, but after all, there were no such things in Diablo I or Diablo II.
There were some player complaints shortly after health globes were revealed, as some players felt they were cheesy; too reminiscent of "power-ups" as found in fighting games or other RPGs. The developers never wavered in their support of this gameplay mechanic though, and this issue seems to have been put to bed, in terms of player complaints.
The first Health Globe was shown at WWI 2008 with the announcement of Diablo III and the Barbarian/Witch Doctor. It was originally just a floating red "blob", but has since received a graphics update (as of BlizzCon 2008) with a little golden crown on the top and bottom.
Health Globes as well as Mana globes were present in the Blizzcon 2008 demo, where fans first got a chance to play the game.
By BlizzCon 2009, the Mana Globes had been removed and the function of Health Globes had been modified a bit, with a few Wizard skills (such as Stability Control) and Witch Doctor skills (such as Spirit Vessel) modified to allow mana gains from the consumption of health globes.
This changed further in early 2010, when the Wizard's new resource, Instability was revealed.  This resource, later changed to Arcane Power, was not affected by health globes. The Witch Doctor, on the other hand, continued to gain mana from health globes.
- Wizard Gameplay Blizzcon 2008
- GamesCom 2009 demo
- Full Wizard Skill Trees, Blizzcon 2009
- Health Systems: Diablo3.com
- Witch Doctor traits