The Battle Arena is an arena where you fight opponents for battle points (BP). It resides in Soul Society. To enter the arena, you must pay a small amount of gold (nothing you can't afford). In these fights, you can fight different leveled opponents, treated just like regular Hollow fights, but they're not Hollows. Your reiatsu is restricted, meaning you must rely purely on your own strength. Opponents start at level 9, but you may fight them at any level (if you're level 10, you can still fight the level 9 opponent). After defeating a few level 9 opponents, you may fight level 10, then level 11, etc (again note that you may fight any level opponent at any level). The higher the level of the opponent, the more battle points, and gold, you will ultimately receive.
When you defeat an opponent in the Battle Arena, you will receive a small amount of gold (a flawless fight will earn you a little extra gold), but no experience and your hitpoints will be restored back to the amount you had when you entered the Battle Arena.