Clothing Sets are, as the name suggests, sets of clothes that are meant to be worn together. When multiple items in a set are worn, the wearer gets a stat boost.
Wearing individual items in the set will give you that item's bonus (shown in purple).
Wearing multiple items in the set will give you the set bonus, which varies depending how many items in the set are worn. These bonuses are shown in grey when they are idle, and green when they are active.
They are all found in dungeons, except for the Role Upgrade set, which is found by completing Roads to the Dragon quests.
For a list of the clothing sets, see Clothing Sets