It will depend on the language/database that you have available to you on your server. I know PHP/mySQL forums best so here are a few I have experience with:
Invision Power Board - http://www.invisionpower.com
vBulletin - http://www.vbulletin.com
phpBB - http://www.phpbb.com (Open Source)
Simple Machines - http://www.simplemachines.org (Closed Source)
myBB - http://www.mybboard.net/ (Open Source)
Personally speaking on my sites I use Invision Power Board. As far as free ones go I would go simple machines over phpBB for ease of use, even though it is a closed source project.