phpBB , Smf are the free 1's and both are good

I prefer phpBB though

I'be used a number of them, it really depends upon the level of integration you're trying to achieve.

Today I have spent some time with Simple Machines Forum (SMF as pfjsolutions mentioned). I chose it for my current project over PHP BB because it is easier to integrate with my user database. In my application I want members of my site to automatically become members of the forum without having to register or log in separately. SMF has an API that should make this easy (I say "should" because I'm just learning about it right now, in about 3 hours I'll be an expert!).

Both of them have a number of available themes that can make customizing them easy. They are both very easy to install as well.

Bill

PHPBB is the most community supported open-source forum software based on PHP/MySql. It has rich functionality and easy to use. You can search Google with "PhpBB demo" and try the demo installation before you install it to see if it fits your needs(The software will be re-installed every 1 or 2 hours to make it like new install.)

You can try vBulletin too. Its paid but most big forums use vBulletin.

Yes right these day you will find most of the forums implemented either in phpbb or
vbulletin platform.phpBB is the free forum hence the ideal choice for most of the users,including me.so what you can do is start with phpBB,once you make some money and if there is a need and desire could invest into vbulletin.
All the Best!!!!

For free forum softwares I'd go with phpBB3 and SMF.

SMF is better if you're using joomla for your main site, but phpBB is better if you use a wordpress blog. If you don't plan of integrating them it doesn't matter. And or if you have a static site or use a different CMS you shouldn't worry about which one you choose.

Personally I'd go with phpBB3 regardless simply because it simple to use and it's code makes since. Edits are easy and you can modify the themes (like add a link back to your main site) buy just pasting the code into the theme. I've use SMF but just found it the admin panel too busy and simple tings were made more complicated than neccissary.

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