Diablo 3's only way to battle other players is in the Arena, as opposed to Diablo 2 where you could fight players in the regular game. The Diablo 3 PvP Arena was first announced a Blizzcon 2010 with the following video:
Before we get started on how exactly the Arena will work, it is good to know that the Arena won't be in the initial release of the game, but will come in a later patch. Blizzard announced this on the 9th Match 2012:
As we're counting down the days until we're ready to announce a release date for Diablo III, we've come to realize that the PvP game and systems aren't yet living up to our standards. Today, we wanted to let you know that we've made the difficult decision to hold back the PvP Arena system and release it in a patch following the game's launch.
As stated many times by Blizzard, Diablo 3's PvE and PvP will be completely seperate. You can be sure that no balancing change for PvP will ever effect PvE or viceversa, like in World of Warcraft. This could potentially mean that many Diablo 3 PvP skills will be differenet than their PvE counterparts.
As you can assume, Arenas will contain no monsters or NPC merchants, and drop no items. To get gear for the PvP Arena, you have to kill monsters (remember that Diablo 3 is a PvE oriented game). The Arena will still reward players with Achievements and Rankings (and bragging rights!), and maybe other things that have not yet been announced.
One thing that throws me off from looking forward to the Diablo 3 PvP is that you can't have a "Dual Spec" character. "Dual Spec" in other games allows you to have 2 versions of your character, one for PvP and one for PvE. This means you can be both effective in PvP and PvE and not having to choose which path to master. Blizzard confirmed this in the following quote:
Hi, do you have any plans on allowing the items, traits, skills etc. used in PvM to be swapped for a PvP build on the same character?
--Nielsdue No current plans for a dual spec or other quick-swap mechanic to change into a PvP build. --Diablo
A potential fix to this would be to level a character to the desired level, then respec him to a PvP oriented build and stick with it. Still this is more trouble than most will want to go thru to PvP and many will simply choose just play PvE only or PvP only.
Apart from that, the Diablo 3 PvP Arena will feature matchmaking and ladders, and a "best of" mechanic taking into account all the rounds in the match. Currently only the 3 v 3 mode has been announced, but other sizes may also be supported like 1 v 1. As for game mode, peeking into the beta client files reveals DoTA and free for all game modes, but there is no guarantee that this will be in the final release.
There will be many different Arena maps where players can duel, each with their own layout to favor different situations. For example a map with pillars and other obstructions will hinder ranged characters, while a map with many pits and low debris will favor them.
There has been no entry requirement set yet for the Arena, but you can be sure that the real ladders will be for level 60 characters.
Finally, as mentioned in our Diablo 3 Hardcore post, there will be a Hardcore-only PvP arena will NOT have permanent death.