Arena
Arenas are places where teams of players can compete against each other in deathmatch-style PvP. Instead of participating in objective-based PvP, these contests will center around your team's ability to obliterate the other team.
Battlegrounds
Battlegrounds are instanced areas solely used for the purposes of PvP. Each BG has two red instance entrance portals, one for Alliance, and one for the Horde. They attempt to allow balanced numbers into each instance, however, this is not always the case.