After making your favicon.ico image and placing it in the web root - public_html directory - of your site(s), you then have to add some code between the <head></head> section of your web site's index file - index.html perhaps, usually after the meta tags.
Best to put the file in the web root for consistency - there are also new tags that you may wish to include which relate to other browsers and can be found from the below Google search.
Code:
<link rel="shortcut icon" href="favicon.ico">
Google for
favicon.
If you have more than 1 site hosted - addon domain or parked domain for example - in the same web root - public-html directory - then repeat the above procedure with a different favicon.ico file for each, if required, in that particular web site's sub directory.
Regards,
Rocky