Is there PvP in Diablo 3? How does it work?

Pvp was added in patch 1.0.7. It is called Brawling. For details click the link Brawling.

You can brawl with other players in your party, either one on one or in a group free-for-all, to prove your dominance. There are no rules, scores, or limits.

When you want to start something, talk to Nek the Brawler. You'll find him in each act’s town area, near the waypoint.

Nek will transport you to the ideal environment for a fight – The Scorched Chapel, a flat land that includes flowing waters, a brick church, and a graveyard with room for new arrivals. Any members of your party can join you there by speaking to Nek. Brawling is opt-in – you won’t get attacked by anyone who isn’t in a group with you and currently present in the Scorched Chapel.

When you die in the Chapel, you’ll reappear there (even if you’re a hardcore character, your death isn’t permanent). You won’t lose any equipment durability for participating (or perishing) in a brawl.

PvP will not be available immediately; Blizzard has said this will be coming in a later patch so they can get the game out to us hungry people sooner.

http://us.battle.net/d3/en/blog/4574895

Diablo III PvP Update

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 for how it will look when done:

When the PvP patch is ultimately ready, it will add multiple Arena maps with themed locations and layouts, PvP-centric achievements, and a matchmaking system that will help you and your team get into fairly matched games quickly and easily.