Ah, so there are a couple of things that you would need to do.
First is to migrate the database and entire board over to the new server. Once that is done, you'll need to migrate the DNS (by logging into your registrar and changing the name servers).
During the DNS migration however, there is a very real possibility that some users will go to the old site, and others to the new site. I'd strongly suggest implementing some method on the old site to either 1) Let users know that the site has moved and provide a link or 2) Redirect them automatically.