Some of the most popular forum softwares:
Invision Power Board
A commercial forum hosting from the creators of the popular Invision Power Board forum software.
Simple Machines Forum
A free community software package with themes, fast message database, secure file-attaching system, and mod installation manager.
vBulletin
vBulletin (vB) is a commercial Internet forum software produced by Jelsoft Enterprises and vBulletin Solutions, both subsidiaries of Internet Brands. It is written in PHP and uses a MySQL database server.
phpBB
PhpBB is a popular Internet forum package written in the PHP scripting language. The name “PhpBB” is an abbreviation of PHP Bulletin Board. Available under the GNU General Public License, PhpBB is free software.
Vanilla
Vanilla is an open-source, standards-compliant, multi-lingual, fully extensible discussion forum for the web.