For example, in Appointments:
I am going to be out at a doctors appointment and would like to track the location and time in my calendar, but don't necessarily need everyone to know the specifics. Selecting a Type of "Private" for this Appointment will still block out the time, as other users can see in the Attendees tab, but will restrict actually opening and seeing any of the details to myself.
Another example, in Contacts:
I would, likewise, prefer to keep my Doctor's name and contact information in the system for convenience, but don't need everyone else to see him at all. I check the box labeled "Private" on the Contact entry and only I will see that person in my list and be able to view or edit the entry.