The Battle Square in the Golden Saucer is probably the most rewarding mini-games in FF7. After paying a 10GP registration fee (you can earn GP at Wonder Square and the Chocobo Races) you pick a single character and head of into the arena. There, you fight a gauntlet of up to 8 enemies with increasing difficulty levels. On top of this after each round, if you choose to carry on fighting, you are given a handicap (see handicaps chart) which can make what would normally be an easy battle into an impossible task. After you have completed 8 battles or if you choose to quit between rounds you will receive BP which can be exchanged for prizes, the more severe the handicaps you suffered then the more BP you earn. You will also gain a consolation prize, a "tissue". If, however, you die you will receive none of the BP you earned. Any changes to your HP, MP, Limit Meter, Status, etc. will not carry outside the battle arena, so even if you die you will appear outside with the exact same stats as you came in with. Similiarly, you can gain no EXP, AP or Gil in the battle arena.
Strategy
If you're going for the big prizes (Omnislash, W-Summon) it is almost universally accepted that you should wait until disc 2 and 3; when the prices are lower and until you're around level 60; so you stand a chance.
It is probably a good idea to do this with Cloud, simply because he will always be your strongest character. If you have them, equip him with Ultima Weapon and a Mystile, if not just find the strongest weapon and armour you have and equip that: don't worry about materia slots (the Ziedrich can be stolen early on from Rude in Gongaga and is a good choice for this). Also, try and get your limit meter up to about 4/5 of the way before you go in (it will then be at 4/5 every time you enter the arena). A ribbon is essential. This done, go to the Golden Saucer and get about 100 gp, I did my time in the battle arena immediately after breeding a golden chocobo and so had plenty of GP from racing, however it shouldn't be a problem to get enough GP in Wonder Square later on the game (when the Sub and Snowboard games are unlocked).
You're now ready to fight, remember that you will lose your BP if you leave Battle Square so this all has to be done in one go. Equip Cloud with as many Counter Attacks, HP Pluses and MP Pluses you have as well as an Enemy Skill with Big Guard and most importantly Slash All (see the materia section and enemy skill section for help with these). When you get inside the arena cast Big Guard straight away and start attacking. By the end you might get into trouble because of handicaps, a broken accessory or weapon can be a thorn in your side so if you find you just aren't doing enough damage raid your inventory for those items you've accumulated in the course of the game, or use Enemy Skill if it isn't broken by now. Heal any status effects you can with items. Use this strategy to build up 32 000 BP and purchase Omnislash. To maximise BP, try and get All Materia broken before the last round, this isn't as bad as it seems when you are mainly using physical attacks and items, Slash All will be disabled but that doesn't matter as there are no groups beyond about round 6.
Once you have Omnislash the Battle Square is easy. Spend any leftover BP on small items and head outside the Golden Saucer, build up your limit meter until you are ready to use Omnislash and then return to Battle Square. This time, all you need is Enemy Skill and Mime, fill whatever space you have left with HP and MP Pluses. Whatever you do, don't equip Counter Attack. In the first battle, cast Big Guard on yourself and then use Omnislash. From then on you can just keep miming Omnislash and wiping everyone out. Hopefully, you have 9999 HP so you should never need to heal, if you do, hold out until the last minute and use an X-Potion - you'll have to use normal attacks after that. Omnislash, at full power, will kill all the enemies in the Battle Arena in one go, if, however, you're weapon gets broken multiple times the damaged can be severely reduced, just keep at it though. You can easily build up enough BP this way to buy W-Summon and anything else you want.
A Dame to Kill For
The woman outside the Battle Square says she likes Fighters, but not those who are all talk. To prove you ain't all talk go into the arena and do a full gauntlet of eight enemies, if you talk to the woman again, she will give you something.
Special Battle
If you have Omnislash, W-Summon and Ultima Weapon the woman at the counter will offer Cloud a place in a special battle. This isn't any harder than a normal Battle Square gauntlet using my Omnislash technique mentioned earlier. Once you win this battle you will receive the extremely handy "Final Attack" materia.