Speak to Shinra on the airship to enter in the fiend arena and enter yourself in a colosseum cup.

Cup	How to unlock
Aeon Cup	Requires capturing and training a specific succession of demonic creatures.
Almighty Shinra Cup (temporarily replaces Fiend World Cup)	Complete Omega Weapon's Fiend Tale.
Cactuar Cup	After beating Grand Cup: Hard thrice (3 times), beating it once again (while defeating the Cactuar team).
Chocobo Cup	After beating Grand Cup: Hard once, beat it once again (while defeating the Chocobo team).
Farplane Cup	Beat every other tournament once (not including Almighty Shinra Cup)
Fiend World Cup	Requires capturing and training a specific succession of demonic creatures.
Grand Cup	Beat Standard Cup: Hard, once
Grand Cup: Hard	Beat Grand Cup, thrice (3 times).
Standard Cup: Hard	Beat Standard Cup, thrice (3 times).
Youth League Tournament	Beat Grand Cup: Hard, 6 times.