1. bbPress
http://bbpress.org/
Pound for pound bbPress is your best option for integrating a full fledged forum into your blog. bbPress has a very simple interface which makes it very popular amongst it’s users. The templates are easy to customize. In fact they are just as easy to customize as Wordpress templates. And the plugin actually integrates the forum database with your Wordpress database, that way the integration is seamless.
2. Forum Server
http://wordpress.org/extend/plugins/forum-server/
Forum Server is a plugin for Wordpress that is a more advanced and up-to-date plugin of the original WP-Forum plugin. It will integrate with your Wordpress Users database table so that users who are registered with your blog will also be registered with the forum. The plugin is a little bit more complex to use than bbPress, but you will find a few different characteristics of this forum that you might like and choose over bbPress.
3. Simple: Press Forum
http://simplepressforum.com/
If you like lots of icons in your forums than you will enjoy this plugin. Again this one is not as easy to use as bbPress. But for those who are accustomed to more traditional looking and functioning forums than this will be a great choice for you. This plugin can too integrate with Wordpress user registrations and logins. This forum appears to have the most overall features than any of the other Wordpress forum plugins. If it is a more advanced forum you are after, then go with Press Forum.
4. phpBB Recent Topics
http://wordpress.org/extend/plugins/phpbb-recent-topics/
There may be a situation where it’s best to install your forum with an open source application, separate from the blog entirely. phpBB is by far the most popular and notable of the open source forum apps. This plugin for Wordpress called phpBB Recent Topics will allow you to include in your blog (anywhere of your choice) a list of the most recent threads or topics from your phpBB forum! This is the quickest and least painful way of integrating a forum with your blog.
5. Wp2BB
http://wordpress.org/extend/plugins/wp2bb/
Wp2BB provides another way for you to integrate your phpBB forum with your Wordpress blog. What this plugin does is that it automatically creates new topics in your phpBB forum for every new post that you write in your Wordpress blog. It can also coexist with the Wordpress comments system or replace it completely. In a nutshell this plugin would increase the conversation in your forum rather than where normally takes place in your blog posts comment threads.
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