For background information you can find out What is a Favicon and How to Create a Favicon in seperate articles, also if you are looking for Adding a Favicon to your Website that is also covered. 

On with the Joomla! Favicon information

In Joomla! 1.0, the favicon was stored in the /images directory of the site. The system automatically loads a reference to /images/favicon.ico in the header of a Joomla! 1.0 site, so you could change your favicon by simply replacing that file on your server.

Of course with updates and new flexibility thing were bound to become slightly more complex and Joomla! 1.5 is no exception to the rule as Joomla! 1.5 handles favicons differently. Instead of having a central location for the favicon, Joomla! 1.5 loads the favicon from the root directory of whichever template is loaded on your site.

For example, the template on this site is located at http://www.afieryphoenix.com/templates/templatename/, so the favicon for this site is located at http://www.afieryphoenix.com/templates/templatename/favicon.ico. If I wanted to change the favicon on this site, I would simply replace that file with a new file called favicon.ico.  This is easily done via your FTP program of chioce.

Usually template designers will either include the default Joomla! favicon in their templates, although some may include their own favicon. It is up to you to take the step of modifying your favicon to suit your own site content or logo.

Sometimes you may not be able to see your new favicon straight away - Do Not Panic! The reason for this is that the default icon is probably still in your browser's cache. If you clear your browser's cache, you will be able to see the new favicon, give Ctrl F5 a go to force a complete re-fresh.

The favicon is a small thing that fundementally does not affect the performance, usability or look of your site, but paying attention to little details like these will give your site a more finished, professional feel.

Now I just need to go and change mine ;0)