This specific search string is a "Google Dork." Here is what each part does:
: This filters the results for folders that have been named "private" by the administrator. intitle index of private updated
By default, most web servers (like Apache or Nginx) are configured to display a plain list of files if there is no "index.html" or "index.php" file in a folder. This is called a . This specific search string is a "Google Dork