I spent a bit of time a few years back now as an administrator over at iKonboard (
www.ikonboard.com), not long after Matt Mecham and his team left to create Invision Power Board (what a success that's been ... great product!). iKonboard being a product which should be considered Dead but for Perl enthusiasts and modders to be honest ... the ownership debacle that has plagued it for years has given it a death sentence.
Realistically your looking at a PHP board, the perl offerings nowadays are not quite as quick to update as most of the PHP solutions so that, in my opinion is your first decision.
Your next decision is cost, free or paid? The options are quite clear.
Paid
Invision Power Board -
http://www.invisionboard.com
Reasonable cost, great feature set with a comprehensive suite of "add ons" available which make your forum more of a website ... good support with their own Hosted Solutions available make them an all round good choice if your willing to stump up the cash for it.
Not had any experience with styling/customization in the latest releases.
Quick to stomp on bugs with regular updates and feedback actively taken on board.
vBulletin -
http://www.vbulletin.com
Sort of similar business model to IPB, before the prior came along they were probably the only paid solution worth consideration - used by NFL teams and various other large corporations the world over are proof of its pedigree.
Not had any experience with styling/customization in the latest releases.
Similarly quick response to bugs and issues raised by users with excellent support.
Free
phpBB -
www.phpbb.com
Possibly the most popular free forum software around at the moment, simple to setup but the Registration authentication is woeful, spam registrations on most Vanilla phpBB installs are common. Styling is relatively easy to do with decent guides available in their forums.
KittenAuth is a relatively new "mod" which sorts out the registration authentication issues - however phpBB 3 is just around the corner and may be worth the wait ...
SMF -
www.simplemachines.org
Good although less well known forum software - brilliant mod system which allows for
very easy installation of addons. Ties in quite well with Mambo and other CMS solutions to give a decent well rounded website.
Styling I found to be a bit of a pain in the rear end, heavy reliance on CSS and if your not completely comfortable with it you may find its User Interface a bit odd in places.
Hope this helps ... take a good look around and look at the example forums on each providers site before making your final choice.
Other things to consider are Moderation and Administration tools - people posting on your forum is not the be all and end all - good moderation is essential, the tools provided to you can make it easier.