Coquettish describes a woman's flirtatious manner. If you were acting in a coquettish way, you might banter playfully and bat your eyelashes.
Coquettish is the adjective form of the noun coquette, a young, flirtatious woman. Coquette is the feminine diminutive of the French word coq, which means cockerel or rooster. The word was originally applied to men in the form of coquet to express how men would strut to attract women the same way roosters do to attract hens. Using coquet for a flirtatious male is now rare, but the feminine version remains.