Now here I can give you plenty of advice.
Probards:
Very simplistic boards but allows probably the most wide variety of themes that I have seen in all of my life of coding. There are tons and tons of users for these free self-hosted boards. We used to run these in school because our system admin failed to block them and talk shit about teachers all day. Truly one of the best free boards, but you'll have some wandering eyes if you're looking for global permission sets and setting up the category/board over and over is more time-consuming than the other types of boards. I would say if you're looking for true customization but within a specific set of rules, Proboards is the way to go.
PHPBB2:
Plenty of Mods/Themes/Packages that are free to install and overall probably the most customization. You will never not be able to find what you're looking for in terms of Shops/RPGs/Ranks/Usergroups. However, the drawback is that PHPBB2 takes a lot, a lot, a lot of customization before it actually is comparable to modern day forums like VBulletin/Invision Power Boards. PHPBB2 was one of the key leaders back in the day, but I'm afraid that PHPBB3 is now the focus of the development. I would only recommend PHPBB2 if you have used them before and can't get used to PHPBB3.
PHPBB3:
A newer software, fresh off the chopping block and tons of added capability from PHPBB2. Custom Usergroups and global permission sets are available from the get-go. Also free but you may have a harder time finding pre-made themes for this software since it is so new. At the same time, this is probably the closest you'll find to a free VBulletin and the support will only grow over time. If you are a developer of such graphics, then I'd recommend this to you, but if you aren't a big graphic buff you may not like this as much as some of the others. Check out SMF if you are considering PHPBB3, as it may save you a lot of time searching for themes/customization.
SMF:
Smart Forums. I never really thought I'd leave my PHPBB-based boards until I saw these free forums that you can host on your own server. This is the best of both worlds in terms of PHPBB vs. IPB (Invision Power Boards). It's an open-source forum which means anyone can be a developer and post their submissions on the SMF boards. The drawback is that when it comes to having your site be 'original' you may have issues finding themes that aren't overused on other SMF boards. However, if you don't really care about the originality aspect, I'd say this is the best overall free board.
VBulletin and IPB are nice, but I would never pay for the heavy-costing licensing fees for the reasons I was hosting these forums back in the day. Basically only guild-forums for WoW and other games I played.
Here are some links to some free already hosted versions of these softwares so you don't have to go through installing them, and can simply just try them:
http://www.proboards.com
http://www.phpbber.com
http://www.freephpbb3.com/home.php
http://www.simplemachines.org/
http://www.invisionfree.com
http://www.vbulletin.com/admindemo.php
Good luck and if you decide on Proboards, SMF, or PHPBB2, I can probably help you with any questions you may have via PM. Good luck!
