An Internet forum, or message board, is an online discussion site. It is the modern equivalent of a traditional bulletin board, and a technological evolution of the dial-up bulletin board system. From a technological standpoint, forums or boards are web applications managing user-generated content.
People participating in an Internet forum can build bonds with each other and interest groups will easily form around a topic's discussion, subjects dealt within or around sections in the forum.
http://en.wikipedia.org/wiki/Internet_forum

MyBB is a powerful, efficient and free forum package developed in PHP and MySQL. MyBB has been designed with the end users in mind, you and your subscribers. Full control over your discussion system is presented right at the tip of your fingers.

phpBB is a high powered, fully scalable, and highly customizable Open Source bulletin board package. phpBB has a user-friendly interface, simple and straightforward administration panel, and helpful FAQ.

Elegant. Effective. Powerful. Free. SMF is all of the above. SMF is a next-generation community software package and is jam-packed with features, while at the same time having a minimal impact on resources. And, yes, it is free.

PunBB is a fast and lightweight PHP-powered discussion board. It is released under the GNU General Public License. Its primary goals are to be faster, smaller and less graphically intensive as compared to other discussion boards.

With Vanilla we wanted to break the mold created and followed by just about every other forum on the web. We sat down and thought about what we liked, and more importantly, what we didn't like about web forums.

Phorum is a web based message board written in PHP. Phorum is designed with high-availability and visitor ease of use in mind. Features such as mailing list integration, easy customization and simple installation make Phorum a powerful add-in to any...

UseBB is a light and Open Source PHP and MySQL based forum package, distributed freely under the GPL license. Developed with simplicity and usability in mind, UseBB is ideal for small to medium sized websites which need a clear and efficient forum package

XMB is a lightweight PHP forum software with all the features you need to support a growing community. With outstanding community support and contribution, you will find XMB to be easy to setup, customize, and enhance.

my little forum is a simple PHP and MySQL based web-forum that displays the messages in classical threaded view. It's Open Source licensed under the GNU General Public License. The main claim of this forum script is simplicity.

PBBoard: Power bulletin board.
Simple, Fast, and very secure Open source php program.

FUDforum is a super fast and scalable discussion forum. It is highly customizable and supports unlimited members, forums, posts, topics, polls & attachments. It can import XML Feeds and sync with USENET groups and Mailing Lists (bi-directional).

Viscacha is a free bulletin board system with an integrated CMS. The intention of the software engineers is to combine the current standards with new and user friendly features. The system supports packages for easily extending the core system.

OmegaBB brings AJAX thinking to forum software, making a realtime forum unlike all others. Feature-rich, with over 100 configuration options and a plain-spoken source code, it is a platform ready to be customized to your liking.

Elite Bulletin Board is an advanced Bulletin Board program that provides advanced features such as CAPTCHA, sub-board, skinning ability, multilingual, commercial password encryption, and much more.

Dove Forums is a Open Source Discussion Forums Software built on the Codeigniter framework. We built it for speed, simplicity and usability.

KaiBB is a free bulletin board system written in php and powered by a mysql database.
Designed to be the balance of easy to use and powerful, this software has everything you need to run your own forum based website.