A few years back my favorite was Invision, but that is no longer free and I have not used it since the free version was around.
I'd have to say the "standard" for open-source forums is PHPBB and it does a pretty good job. However, vBulletin seems to be the best when it comes to large scale communities, although its not a freebie.
PhpBB is always in my top list second is smf but this days the smf is not having enough functions and the installation of mod functions required skill.
SMF is hacked by the hackers several times when i used but phpbbis safe in this and i prefer phpbb
PhpBB and then smf forum will be the best choice than other forum softwares, the recent version of smf is some what unstable try the older version, it have lot of mod functions.
I've used PHPBB many times and found it very user friendly and easy to use. I also recommend VBulleting like other users have as it is an excellent forum script. Anybody remember UBB? Ultimate Bulletin Board. . . Back in the day this was the best script in my opinion. http://en.wikipedia.org/wiki/Ultimate_Bulletin_Board