If you don't want the /forum in the URL, then yeah, do something like:

cd [webroot directory]
cd .. #go up one directory
mv htdocs htdocs-old
mv htdocs-old/forum htdocs

If you have a lot of absolute links (i.e., href="/forum...") that break, put things back where they were, and do some URL rewriting, maybe something similar to this:

<IfModule mod_rewrite.c>
RewriteEngine On
RewriteRule ^/index.html$ /forum/ [R]
RedirectMatch ^/$ http://mywebsite.com/forum/
RedirectMatch ^/index.html$ http://mywebsite.com/forum/
RewriteBase /forum/
RewriteCond %{REQUESTFILENAME} !-f
RewriteCond %{REQUESTFILENAME} !-d
RewriteRule . /forum/index.php [L]

You might need to enable mod_rewrite first to make it work as expected.