Player versus Player
From Diablo Wiki
PvP, or Player versus Player is head-to-head mortal combat, pitting players against other players. The design focus of Diablo III's multiplayer is on PvE cooperative play, but PvP is supported and was introduced with Patch 1.0.7.
To brawl with other players, speak with Nek the Brawler, who is located near the followers in each act. Speaking with Nek will take the players to the Scorched Chapel, a zone designed specifically for PvP. It has custom geometry and its environmental features can be used in combat.
Brawling currently supports up to four players in a Free-For-All format, which means players can battle their friends either one-on-one, three-player FFA, or a full four-player FFA. If a player dies in the brawling world, they can either be resurrected by another player or they can respawn to a random location in the Scorched Chapel. Death is not permanent in the brawling zone, even for Hardcore characters. There is no durability damage to items while in the brawling zone. When brawling is over, players take a portal out or use a Town Portal to return to town.
The game does not keep score and there are no objectives. However, a player's death is announced using yellow text in the chat log, which tells who died and who killed them. The system is inchoate and there is no official scoring or ladder system, but there are third party websites that provide rankings such as Diablo3Ladders.Com. A public game search option became available with patch 1.0.8. It is not known if there will be any achievements awarded for PvP in the future. Balancing is expected to be finished with the release of patch 1.1.0.
History of PvP in Diablo III
It was originally planned that Diablo III would ship with PvP but as time crept on Blizzard took the decision to launch without as the PvP element was not meeting their expectations and to get it right would have delayed the release of Diablo III further. They felt it could launch without PvP. 
PvP is due to be introduced with Patch 1.1
And when it does launch, Blizzard have confirmed there will be no PvP specific gear, found or crafted 
In December of 2012 Blizzard announced that the PvP element of Diablo III had been simplified with the Battle Arena aspect of the game going to back to the drawing board. A more simplified version of PvP was promised for Patch 1.0.7. 
PvP Development in Diablo III
Unlike in Diablo II, there will not be any non-consensual PvP in Diablo III; there is no more "hostile switch" or any other way to duel or PK in a normal game. PvP will be supported only in special dueling Arena games.
- Jay Wilson: We'd like there to be a dedicated PvP mode, and we'd like to move away from [how it worked in previous Diablo games] where players just enabled PvP. We don't have any specific plans yet because we haven't really made any active decisions. The only real PvP-oriented decision that we've made and announced is that we do not allow the that Diablo II had where you can go into town, go hostile, pop back through a town portal, and insta-kill your friend. That just makes people not want to play the game. I know some people say, "Oh, you're taking the teeth out of Diablo." I understand why they feel that way, but making people not want to play together does not make for a better game. That's our feeling.
- We definitely want there to be a PvP mode for PvP players, and we would like that mode to be a really serious, skill-based, very strong [aspect of the game]. I feel that on the side of all our games, we really try to make PvP games that cater to a competitive player, first and foremost. We don't try to dumb down or tone down our PvP games. We make [them] good, strong competitive games. StarCraft is one of the best examples. But in terms of what our actual plans are for Diablo III, we don't have anything to specifically announce right now, mostly because we're still messing around with a bunch of different ideas.
The philosophy isn't as much to remove pvp as it is to compartmentalize it into a specific dueling arena. The development team, as shown above, views PK'ing as another form of griefing, particularly with the implication of exploits such as tppk.
PvP Does Not Slow Game Development
Spurred by a fan who asked if Blizzard could leave the PvP out of Diablo 3 at launch, and patch it in later in order to release the game more quickly, Diablo 3 Community Manager Bashiok explained why that's not an option and why PvP doesn't add much time to the overall development. Here's his quote, from May 2010. 
Diablo III did not ship with a Player Vs. Player game mode.
- PvP not ready for launch of Diablo III - Community Manager, Blizzard, 12/10/2012
- PvP in Patch 1.1 - Community Manager, Blizzard, 20/09/2012
- PvP Specific Gear - Jay Wilsion, Blizzard, 01/12/2012
- Future of PvP - Jay Wilson, Blizzard. 27 December 2012
- Patch 1.0.7 Preview - Jay Wilson, Blizzard. 12 January 20132
- Development Teams' philosophy towards PvP - Jay Wilson, 20/08/2008
- PvE, PvP simultaneously developed - Community Manager, Blizzard, 06/05/2010