Every thief can change their outfit in the dungeon. Tap on your throne and open Outfit tab (shoulder icon). There you will see all outfits you can wear.

Unlike thrones, outfits don't give you any additional perks and are mere for aesthetic pleasure.
To unlock an outfit you need to either spend some orbs or use a Style-O-Thief machine.

Style-O-Thief is a special mechanism which allows you to get a new outfit by spending magical ingredients. Each added ingredient fills up the progress bar of the machine. The rarer the ingredient is, the faster it fills the progress bar. Fully charged Style-O-Thief machine gives you a better chance to win a rare outfit.

If you happen to win an outfit which you already have, you will get orbs instead of it.

Please note, that some outfits can't be opened with orbs or in Style-O-Thief machine. You can only get them for an achievement (like King of Kings outfit) or as a present (like Christmas costume).