Public Report:
The report will be available to all users who have the report enabled in their permissions. If you select this option, enable the check box next to each permission profile you want the report to be accessible to. To make the report available within all of the permission profiles, choose Select All.

Private Report:
Report will only be available to the user who created it. The Private Report option only displays if the user has Add/Delete Private Reports permission.

Note:
You can specify a report as public or private only when creating the report for the first time. You cannot change this setting after the report is created unless you save the same report under a new name.