The premier PvP arena in destiny 2 is called The Crucible, you will not be able to access this feature until you have completed a few campaign missions. The Crucible is not just a PvP contest but also a way for your Guardians to train and earn better equipment.
Playing The Crucible Matches will earn you Milestone rewards for participating. Even if you don't win you will still earn rewards for just playing. Rewards are given for every match type and joining the competitive Playlist can get you even more. The Quickplay option gives up lots of Engrams and the same rewards can be unlocked whether you play Competitive or Quickplay as they draw from the same loot pool.
Below is a list of the maps you will encouter in the Crucible:
The Dead Cliffs
Location: European Dead Zone, Earth
Modes: Quick Play & Competitive
Endless Vale
Location: Arcadian Valley, Nessus
Mode: Quick Play
The Fortress
Location: Eurpean Dead Zone, Earth
Modes: Quick Play & Competitive
Legion's Gulch
Location: European Dead Zone, Earth
Mode: Quick Play
Altar of Flame
Location: Caloris Basin, Mercury
Modes: Quick Play & Competitive
Vostok
Location: Felwinter Peak, Earth
Modes: Quick Play & Competitive
Midtown
Location: The Last City, Earth
Modes: Quick Play & Competitive
Javelin-4
Warsat Launch Facility Lo
Modes: Quick Play & Competitive