What is the most popular open source message board software?  I want
to integrate the message board functionality with my existing system
such that every time someone posts a message it adds the screen name I
have for that user in my system.

mprrish,
    I don't think you're going to find an exact answer to the question
of the most popular open-source message board software.  However, in a
2004 interview with the founder, phpBB was described as 'one of the
most popular boards on the Internet, powering communities of thousands
of sites'
http://www.sitepoint.com/article/james-atkinson-founder-phpbb

phpBB is open source and available here: http://www.phpbb.com/

With the limited information you've given though, you might find it
useful to look through this extensive directory of other message board
software, and find something a bit more tailored to your specific
needs.
http://www.thinkofit.com/webconf/forumsoft.htm

You also might find this question will get answered sooner if you
narrow things down a bit, such as defining the scripting language you
wish to use.

I hope this helps.