Leaderboards let you see the scores of other players. These scores are based on the song, the instrument used, and the difficulty.
You can access the leaderboards by heading over to the organ in the middle of backstage. Select a song and press your leaderboard button. Here, you can compare your scores on the leaderboard between you and your friends or globally, against players around the world. You can even see your scores when you play solo, duo, trios, or quads.
Make sure you fill the Star Meter as much as possible to achieve a high score. Good luck!