Win the Grand Cup once; requires a Chocobo in the party!
Win the Grand Hard Cup thrice; requires a Cactuar in the party!
Youth League Cup
Win the Grand Hard Cup six times
Capture the Flame Dragon (Besaid), the blitzball-shooting thing (Thunder Plains), Jumbo Cactuar (Bikanel), Azi Dahaka (Bevelle), Aranea (Via Infinito 20F), Critical Bug (Bevelle), White Flan (Calm Lands), and Mycotoxin (Zanarkand). They will transform into various Aeons; once you get them all, you will need to release them to unlock the cup.
Demon Cup / Shinra's Cup
Capture a Deava, level it up three times, and release it for the storyline end and its replacement by Aka Manah. Do the same for him, his replacement (Aeshma), and his replacement (Shahi). Now capture a Tomb enemy and level it up three times and release it. Repeat this with its consecutive replacements: Dorumen and Monument and Ultima Weapon. Don't alter their default names. After getting Ultima, there will be some more story to deal with and you'll be able to fight this Cup in the Colosseum. After fighting the cup, defeat Omega Weapon and pick him for your party after the fight. Level him up three times and release him, then Shinra comes along. You can now play the altered version of the Cup, the Shinra's Strongest Creation Cup. Play it and level up and release Shinra as usual and you'll get the Cup back to the Demon's Cup.