When Opera cannot connect to a site, it shows you a page "Network error", as on screenshot. Sometimes that happens because site is down and no other user can connect to the site at this moment. "Up For Everyone" detects network error page and uses web service DownForEveryoneOrJustMe.com to check availability of the site for other users. At the bottom of network error page you will see results of that check.
This extension was inspired by Google Chrome's built-in behaviour.