Battle Square:
The Battle Square is a large arena where you can enter a series of battles to earn prizes.
Each battle will earn you BP (Battle Points) which can be exchanged for prizes (3) but your points are reset if you leave the square.
When you arrive at the Battle Square station, head up the stairs to the main lobby. Once inside the arena lobby, there is a room to the right (1) called "Dio's Show Room" which contains Dio's collection of rare items.
If you talk to the receptionist (2) she will ask if you want to enter the Battle Arena for 10 GP, selecting yes will enter one of your party into a battle.
Each battle is 8 separate battles, back-to-back, between each battle is an option to back out followed by a slot machine with a condition that will affect your next battle.
There is also a Special Battle that you can do after you have purchased Omnislash and W-Summon Materia and Cloud has his Ultima Weapon.
Winning this battle will get you the Final Attack Materia. After the special battle, talk to the woman at the bottom of the stairs in the station to get Sprint Shoes.