How can I add a Forum, Message Board or Discussion Board like vBulletin or PhpBB to WordPress?

What you would have to do is take a Forum like PhpBB’s which is great and I’ve been using for a long time.

There are various services hosting a forum for you for free:
http://www.forumfree.com
http://www.freeforums.org
or any other you find searching on google.
However these services are mostly full of advertising, unreliable and little customizable.

I would really recommend you to install the Forum on your own server. It’s much more reliable and much more customizable. If you don’t have a hosting yet or your hosting doesn’t fit the minimum requirements have a look at the Hosting and FTP applications FAQ.

You can read all you need to know about the installation process of PhpBB and how to customize your Forum at my other website PhpBBFAQ.net

Once you installed your Forum I would recommend you to add a page to your WordPress for it. Then either link to it, use an iFrame to embed the Forum into the page or make a Redirect to it (as I did for my WordPressFAQ.org Forum).

Note: If you want to install a PhpBB Forum on HostExcellence read here