If you are going for the free ones go for Php or simple machines as they are the leading ones in the free forum software world. For paid ones Vbulletin is the best one although there might be quite a few alternate options available for Vbulletins in the paid world.

In my opinion, phpBB is the best free, and vBulletin will always be the best premium option, from experience managing most kinds of forum software (that i know of, anyway).

I like phpBB3 because its free, it has tons of mods, plugins and it will update on its own and even merge the new files with the old automatically.
Its one of the best forums i have tried and i know a website using phpbb that has over 2 million users and they seems to doing great
Demo : free forums