A costume is a piece of equipment that can change the onscreen clothing of any character. Each character has their own unique set of costumes. They have been present since the very first Hyperdimension Neptunia and in every other installment past it.
Since costumes are primarily used for aesthetic reasons, they do not provide any worthwhile additional stats to a character when equipped. In Hyperdimension Neptunia and Hyperdimension Neptunia mk2, all costumes came with an additional stat increase in some area.
However, by Hyperdimension Neptunia Victory nearly no costumes provide any statistical increase to the character outside of special costumes such as DLC ones. This trend would carry on into later titles of the series.