I just recently installed Joomla! using a database that already had an instance of Joomla! installed in it. It was actually really easy. This article explains how to install multiple instances of Joomla! with 1 database.
Update February 7, 2009: I have now written an an article on how to move your Joomla! 1.5 site to a new server.
Things happen. Plans change. Websites outgrow current servers and need new servers. Hosting companies provide poor service, prompting a web master to decide to change servers. Whatever the reason, you might need to move your Joomla! installation from one server to another.
If you've been running a website for a while, you have probably already accumulated quite a bit of content and several 3rd party components and/or modules, and you do not want to start fresh with a new Joomla! installation. This article explains what I did to move HowToJoomla from my test server (howtojoomla.corywebb.com) to the server that it is on now.
Update July 15, 2010: I should have posted this a long time ago, but the BEST way to move your Joomla site to a new server is to use Akeeba Backup.
The most popular article on this site covers How to move your Joomla! site to a new server, but it does not cover all of the details for moving your Joomla! 1.5 site to a new server. This how-to discusses the process of moving your Joomla! 1.5 site to a new server. The information contained in this how-to also happens to be an excerpt from my upcoming book, Beginning Joomla!. (Due out in early April)
In How to Lose the Splash Page, I talk about how to move your Joomla! installation between directories on your server. Because the topic was buried in the "Read more..." section of the post, there may be some readers who ignored it or did not read it. If you want to know how to do this, click on over to How to Lose the Splash Page, and scroll down to "Option 1: How to Move Your Installation".