For free forums there are three biggies(I have tried out all of these in the past few months . I've included my personal experience with them in here):
1. PhpBB:
-very large user base and lots of developers , theme arists , etc.
-very large database of modifications , themes , developer resources , etc.
-Good with security , stability and usability(I have seem some really really massive forums too use it)
-Arguably Most popular of the three
-The newest version(3.06) has really improved a lot and it works beyond my expectations . Now runs with quick reply , and multiple captcha(including reCaptcha) systems , universal user reporting database system , etc. and lots more right out of the box . I currently use this and am highly satisfied .
-Modification installation is to be done manually using ftp and editing of files might be required for certain modifications(unless you are using their automod installer that is also available on their website)
2. SMF
-large user base with many modifications and themes available .
-Lays very high significance on security .
-Large forums like Prize live too use it without many hick ups , so stability is never an issue .
-Oldest of the 3 forums in this list afaik
-mod installation is as simple as browse and install
3. MyBB
-has an uncanny familiarity with vBulletin , some people used to call it the free version of vBulletin
-newest of the 3 listed here .
-mod installation is really very simple .
-the forum is really very light and loads fairly very quickly .
-Lots of hooks and easy to develop(Lots of hooks can sometimes cause security risks if not handled properly)
-Gaining popularity really quickly.
-if you are good with php , this forum provides you all the things you need to customize it to your hearts pleasure . But remember , each customization if not done properly can leave the door open for hackers , so you must be careful with modifications .
There are many-many other free forum systems , but the three above are arguably the most notable of them .
Best Regards,
Aloycasmir