The pedometer counts your steps. To check the number of steps you've taken equip the pedometer and press 'X'. You receive the pedometer from the harvest sprites on your first day.
The pedometer gives the "First Step Note" which is the first note you get. The pedometer also gives you the "10,000 Steps Note" which is received for having 10,000 steps on your pedometer. You have to be carrying the pedometer with you to get this note.