Archness is a quality of being cheeky or playful in a way that's almost rude. If your philosophy professor cracks a lot of jokes during her lecture on ethics, you might comment on her archness.
If you like romance novels simply to make fun of their silly plots and overblown characters, your archness might offend a friend who actually enjoys the books. When you're arch, you tease in a fake or exaggerated way. The noun archness comes from arch, which meant "chief or principal" in the 1500s but grew by the mid-seventeenth century to have a more derogatory meaning after being a familiar prefix for words like rogue and knave.