The Facebook API lets you post status updates (message, links, photos etc) to your timeline via the API. This is handy if you want to have automatic sharing in your application (assuming users have consented to it) to post uploaded photos, links etc.
Facebook, Facebook Graph API, Tutorial
For Halloween, I helped wonga.com reskin an old app they had originally developed with a third-party agency. The app used Adobe Flash as the main user interface, with a PHP / MySQL backend.
Application, Facebook, Wonga.comUse the new Facebook Notifications API to send app notifications to your users, instead of using emails or requests. No additional permissions required.
Facebook, Facebook Graph API, NotificationsAn example of adding Facebook plugins to a YouTube video with some simple styling to make a professional looking page.
Facebook, Facebook Social Plugins, YouTube