One of the thing wordpress lacks is support for community building feature. Forums are strong communities built around specific interests. Many wordpress forum plugins are there to solve this issue.
bbPress – This one is from developer of wordpress itself. This used to be stand alone product but now with release of version 2.0 it has become wordpress forum plugin. It is not as elegant as other forum plugins in functionality and features.
SimplePress (simple-press.com) – It is one of the old wordpress forum plugin developed by two developer. Unfortunately it is not available on wordpress.org plugin directory. It is feature rich and new version 5.0 is under active development.
Mingle Forum – This is another good wordpress forum plugin with great list of features. It is off shoot of WP-Forum. Developer of this forum stopped working on this plugin to work on new wordpress forum plugin. But as it is taking him long, so he is updating this plugin. Future of this plugin will be in limbo.
WP-Forum – This is used to be one of the early forum for wordpress. It has been not updated for last 3-4 years.
WP Forum Server (forumpress.org) – This was forked from WP-Forum. Developer claims it to be advanced and stable. There is a free version and pro version. Main disadvantage with free version is that it does not have SEO friendly URLs. Some developer on wordpress are alleging that it is repackaged version of Mingle Forum. Also plugin there are glitches on their main website, so there is doubt over this plugin.
buddyPress – It is not standalone forum plugin. It is complete social network plugin for wordpress. It does not work with all themes, so you need only buddypress compatible themes to make it work. That is really down side of this plugin. It is yet to be tightly integrated to wordpress and installation and working is not yet smooth for non techie blogger.
There are other hosted forums like vanilla and tal.ki that you can embed in your word press blog. But as these forums are embedded using java script your blog does not get the power of juice.
Some developer are also taking completely new approach to build forum for wordpress using custom post types. See this post for more details.