The role of potions in Diablo III is not yet determined. There are health and mana potions, but no rejuvenation potions have yet been seen or mentioned by the D3 Team. It's not known if there will be other potions such as antidotes, thawing potions, or offensive potions to throw at monsters.
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Potions and Game Balance
One of the D3 Team's main design goals in Diablo III is to make the game more consistently difficult, or at least challenging. Towards that end, they are going to greatly de-emphasize the role of potions in the game, and force players to obtain healing from health globes. It's not known if NPCs will sell potions, or if they will only sell them in limited quantities, but however they are obtained, potions will be less beneficial and less common than they were in Diablo II.
In the Blizzcon demo build (October 2008) there were no NPCs selling potions (or anything else). Healing potions could be found in the demo, but only rarely (characters leveling from 5 to 10 typically found a total of half a dozen potions). They granted a set amount of hit points (40 for a Barbarian at level 8, who had about 150 total hit points), and filled that much health instantly, as rejuvenation potions do in Diablo II.
Potions in PvP
Players have asked how potions will be handled in PvP play in Diablo III. This question is not one the D3 Team can answer very accurately yet, since they've not begun to work on balancing PvP play. However, Jay Wilson was still willing to answer the question during a Q&A session[1] at Blizzcon 2008.
Jay said that potions could not be used that often in D3 PvP, and that there would be some sort of cool down timer on them, as there is in World of Warcraft. They will be useful, but their benefit isn't that large, and they can't be chugged endlessly, as they were in Diablo II. The utility of potions is not designed to be that great, which should help keep them from becoming overly-valuable in PvM or PvP play. Jay thinks that the (relative) absence of potions in Diablo III will improve the PvP play, since potions in D2 essentially gave players infinite health and mana. This turned unregulated duels into little more than potion drinking contests, where the player with more of them usually won.
Elixirs
There are elixirs in Diablo III. So far Strength and Dexterity elixirs have been seen. As of Blizzcon 2008, these provided a temporary, 300 second bonus to that one attribute.
Potions in Diablo II
See the Potion Potions page] in the Diablo II wiki for full details.