If you act coquettishly, you behave in a flirtatious way. Fashion photographers often direct models to pout coquettishly at the camera.
To joke around in a teasing way, or to flirt with someone, is to behave coquettishly. A flirty teenager might smile coquettishly — or your grandmother might joke coquettishly with her favorite waiter at a restaurant. The word comes from its noun form, coquette, "flirtatious person," a French word used in English beginning in the 1660s. In French, a coquet is a male flirt, and a coquette is a flirtatious female.