Phorum - PHP based forum software
Phorum is a message board software that is designed to meet different needs of different web sites while not sacrificing performance or features.
It has hook support which allow webmaster to create modules for using external authentication, altering message data before its stored, adding custom information about users or messages etc. Phorum gives the web designers the power to rearrange the data in any shape, form or fashion they could want. The templates are in pure HTML with simple to understand text commands embedded in them.
It has all kind of features that a forum / discussion board should have. Its features include users and roles management, RSS feed, full text search, private messaging, bread crumb navigation, BBCode, Smiley, Event logging, spam prevention, Developer API support, WYSIWYG editor.
Phorum has more numbers of add-ons like Google-maps integration, Spell checker, Forum stats, Image gallery, User tagging, Highlight search terms, Facebook integration and lot more. External Application could use its User database to provide authentication and user management.
It is widely used and it is used in MySQL community forum. Check out their community page and demo, both are powered by Phorum.