If you say something archly, you express it in a teasing sort of way. When someone acts archly toward you, it might offend you a bit, or it might make you laugh.
You act archly when you put on an affected style of speech, or say something that's playfully sarcastic. You might archly dismiss a teacher's criticism of your work or smile archly as you tease a friend. The adverb comes from the 1500s adjective arch, which originally meant "chief," but was so often used to describe a "knave" or "rogue" that by the 17th century it came to mean "mischievous" or "saucy."