This is a communicaitons system earpiece, custom molded to fit the left ear of Apollo 11 astronaut Buzz Aldrin. It allowed him to easily and comfortably hear transmissions from Mission Control during the mission while he was in the command module.
NASA transferred the communications earpiece to the Museum in 1970 with other Apollo 11 equipment.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.