Costumes can be worn alongside gear to change the appearance of your character, and add stats. There is a button which allows you turn off the costume visual if you wish, as well. You can obtain Costumes ingame through Costume Workshop housing units. Some costumes require Costume Vouchers to make, others do not. More costumes can be purchased via the cash shop as well.
Costume Voucher Exchange
Some clothing sets can be traded in with a costume voucher to receive a costume with the same benefits(or more), allowing you to receive the benefits without changing your gear.
Other costumes can be crafted through Rank 2 Costume Workshops purchasable in Calpheon City. These are Royal Outfits which require a Costume Voucher as an ingredient for workers to create.