Customization is a feature first appearing in Tekken 5, where you can slightly alter the character's looks. In Tekken 6, this feature is greatly expanded upon, allowing to make your character a lot different than they look initially.
A lot of things can be bought, such as items for your character, some of which are used for item moves. Some other things that can be bought are different shirts, pants, shoes and even different hairstyles for your character. Some characters have a third costume, which can also be bought from here. Customizations can be bought with fight money or farmed in Scenario Campaign mode.
In Scenario Campaign, the customization plays a big part, as you can farm items for free and use them to build your character's stats for that mode, such as more health, higher attack power and chances for your attack being elemental.