It depends on the actual forum and what you actually want to do with it. Most major forum software at least as a mod to intergate the userbase with the forum and WordPress.
phpBB
phpBB is known to have quite a few bugs and sadly is quite known to get cracked every other week but updates do roll out pretty quickly. If you want to use phpBB and want to intergate the users, follow this link.
MyBB
MyBB is another free forum software (Not opensource, just free) that is much like the older versions of vB. MyBB support is okay and helpful and someone has made a bridge to intergate the two (Still in Beta, so be careful) and that can be located here.
bbPress
Now I ain’t used bbPress alot but it the feel of bbPress is much like WordPress and offers a bridge to intergate both WordPress and bbPress together.
If I recall correctly, bbPress only works with 2.5.1 correctly right now and an update to bbPress to work with 2.6 however it is still being developed and it is unknown when it is be ready for the public. However, a good point is that it is developed by the same team as WordPress and it is likely they will keep developing it for as long as WordPress lives.
If you want to read the blog post about WordPress 2.6 and bbPress, click here
Plugins
Now, you can also get a forum software that works as a WordPress plugin and the one I would say for you to get is known as Simple:Press. It intergate the userbase so your members do not need to sign up again to access your forum, with some minor editting and look exactly like your site and it can also sync blog posts and forum topics.
I personally would go with Simple:Press as it patches are made quickly, looks lovely and Andy is a nice guy like the support. However, it would to try them all and see which one you like the best.
Just remember, once you have choosen the one you want that it is often hard to then move to another one without alot of hassle. There are many others however this was posted quickly with the free ones I personally use.