You will need to acquire each costume and use its special ability once in battle to unlock the achievement. The costumes, their general locations and the abilities are listed below.
** Any character in the party can equip a costume and execute the moves, it doesn't have to be the party leader and it doesn't need to be done more than once. Thanks to merbear and z for asking.
** Noteworthy is you CAN backtrack to retrieve any pieces you may have missed, prior to beating the final boss.
Robot (start the game with) – Ability: Missile Barrage
Knight (obtained when Everett joins) – Ability: Guardian Shield
Statue of Liberty (obtained by finding three costume pieces in Auburn Pine, design given by kid at the house party) – Ability: Anthem
Space Warrior (obtained by finding three costume pieces in Auburn Haven Mall, design given by security guard) – Ability: Meteor Avalanche
Ninja (obtained by finding three costume pieces in Auburn Haven Mall, design given by group of kids in ninja costumes guarding third level of mall) – Ability: Kitsune Shroud
Unicorn (obtained by finding three costume pieces in Auburn Haven Mall) – Ability: My Pretty Panacea
French Fries (obtained by speaking to the french fry vendor in Auburn Haven Fall Valley) – Ability: Salt Assault
Pumpkin (obtained by speaking to the green glowing scarecrows “Dorrion?” throughout the story starting in the cemetery on the way to Auburn Haven Mall from Auburn Pine) – Ability: All Hallows’ Eve
Vampire (obtained by finding three pieces in Auburn Haven Fall Valley, design given by speaking to the carnival ticket booth vendor) – Ability: There’s a Bat in My Hair
Black Cat (obtained by finding all the Creepy Treat Cards in the game) – Ability: The Cat’s Meow
BiLLzThEd0n has indicated a thread link that can help you obtaining the pieces for the Vampire and Unicorn costumeshttp://www.gamefaqs.com/xbox360/604987-costume-quest/answers...
Please comment if there's anything you feel should be added to help other gamers out and I'll include an edit with credit to you.