BBPress is the official forum script meant to work with Wordpress. It integrates the same user database that controls the logins to your Wordpress blog. So in other words, if people registered an account on your blog to post in the comments section - they can just jump right into to the forums and start participating in the discussions there as well. Some of the downsides to BBPress are that it's not nearly as feature rich and polished as the more prominent forum scripts out there, and visually the layout is pretty rudimentary. The other free and commercial forum scripts won't directly integrate with Wordpress, though some can share the user DB via a plugin or add-on. In terms of visual appearance, it can also be more difficult to make them look integrated with the rest of your site as well. However they are generally fare more robust in terms of functionality. There are free options such as PhpBB, Vanilla Forums, and Simple Machines Forum. There are also paid options such as Xenforo, Invision Power Board and VBulletin. I would say Vanilla Forums is one of the upcoming free options these days, and Xenforo is one of the rising commercial plugins. Some of the older free scripts such as PhpBB are big targets for spam and security vulnerabilities. Some of the more established commercial forum scripts such as VBulletin have drawn some ire over the years for becoming too big/bloated and for charging too much.
