Changing of the Guard
The changing of the guards at the Royal Palace, located in the Old Town (Gamla Stan), happens daily during the summer and is a popular event to watch. The guards actually march, accompanied by a military band, from the Army Museum or Calvary Barracks(when mounted) through the city before arriving at the Palace. At the Palace the ceremony takes approximately 40 minutes. The parade itself takes about half an hour and the ceremony at the Palace starts at 12:15 Monday through Saturday and 1:15 on Sundays and holidays. During the rest of the year the changing of the guards happens on Wednesdays, Saturdays and Sundays.
After watching the changing of the guards, take advantage of being at the Palace and visit one of the many museums located there. Besides visiting the Royal Apartments, you can also visit The Treasury, Gustav III's Museum of Antiquities, and (my favorite) the Royal Armoury. Check the individual websites (linked here) for opening hours and entrance fees. The Stockholm Card gets you in all Palace museums for free.