The Barber Shop is where you can change the appearance of either your Ruler or your various Adventurers after you hire them.
Barber Shop for Your Ruler
You can access the Barber Shop for your Ruler on the overland map by clicking the Kingdom button, then clicking the "Ruler Barbershop" (sic) button.
You can change anything about your Ruler, even their race and gender.
Barber Shop for Adventurers
You can access the Barber Shop for any Adventurer by bringing up that Adventurer's character sheet, then clicking the "Change Appearance" button at the top of the left-most column.Note: the "Beard" button will only function for Dwarf Adventurers.