Each saved report in your database has its own permission settings. If you can't see a report, it means you have no permission to the report. If you can see the report but can't edit it, it means you have "read"-only permission for the report.
The permissions for each saved report are managed from the Advanced Reports page. Users who have "admin" permission to the report are able to edit the report permissions to control other users' permission levels to the report. If a user has the Reports user permission, they'll have Admin access to all reports at all times.
To learn how to update permissions for a report, see How do I edit which users have permissions for a saved report?