Favicon is the favorites icon that is associated with your site and appears in the browser address bar. Both Joomla! 1.0.x and 1.5.x come with a default favicon that displays the Joomla! Logo. You may change that as long as your new favicon is in the ICO format and sized 16×16 pixels. Here's how to do it:
Unlike the 1.0.x series, the only name you are allowed to use for your favicon is favicon.ico but you are offered the flexibility to associate different favicons with different templates. You only need to upload your favicon.ico into the main folders of your front-end and back-end templates, which are found in the /templates/ and the /administrator/templates/ folders respectively, overwriting any favicon files that came with your templates.
However, if you'd rather use a single favicon.ico for all your templates, just upload it into Joomla!'s main folder (that's where your index.php resides) and into the /administrator/ folder. Make sure you delete all favicon.ico files found in the template folders mentioned above because Joomla! will check your template folder first for the favicon.ico file.
To see the new favicon you will need to empty your browser cache.