The Collaborative International Dictionary of English v.0.48:

Flapper \Flap"per\, n.
   1. One who, or that which, flaps.
      [1913 Webster]

   2. See Flipper. "The flapper of a porpoise." --Buckley.
      [1913 Webster]

   3. A flat object used to make a flapping noise by striking
      another object. --RHUD
      [PJC]

   4. A flat and broad object hanging from a larger object,
      either flexible like rubber or hinged to allow a swinging
      motion; a flap.

   5. A young woman who dresses in a modern, stylish manner and
      behaves unconventionally in social situations; -- a term
      used especially to refer to young women during the 1920's
      and their peculiar style of dress.
      [PJC]

   Flapper skate (Zool.), a European skate (Raia
      intermedia).
      [1913 Webster]

WordNet (r) 3.0 (2006):

flapper
    n 1: a young woman in the 1920s who flaunted her unconventional
         conduct and dress