A mouthpiece was part of a woodwind or brass musical instrument through which one's mouth blew air to play the instrument.
In what seemed to be 2383, but was actually 2367, William T. Riker instructed his son "Jean-Luc Riker" in playing the trombone. The boy claimed he wasn't getting a good tone, despite practicing, and Will told him that he was putting too much pressure on the mouthpiece. (TNG: "Future Imperfect")