The best advise I can give is to use a square image as this will give you the best results. So if you have a logo with an image and some writing, use only the image.
Open your favourite graphics program, for example Photoshop. If you don't have Photoshop or a similar program, paint for Windows 7 also does the trick.
Use your graphics program to open your logo.
You now need to resize your logo to 16 x 16 pixels. You can now see why it is best to use a square image!
Save your newly resized logo as a PNG, GIF or JPG image. The best is to save your file as a PNG or GIF with a transparent background, it is not a train smash if you are unable to do this though.
Your logo now needs to be converted to the .ICO format. The are many online favicon generators. My favourite is the Dynamic Drive generator. Go to their website, click Browse and find the resized image on your computer. Then click Create Icon.
Your new favicon will now be created.
Once created, click Download FavIcon and save to your JOOMLA TEMPLATE folder. If your website is already live on a hosted server, save the icon to any location on your PC and upload to your template folder on the server. Example paths are c:\wamp\joomla\templates\YOURTEMPLATE\favicon.ico for a local WAMP installation or public_html/templates/YOURTEMPLATE/favicon.ico if your website is live on a hosted server.
You might still see the old favicon when opening your website, this is because the old favicon is still in your browsers cache. Clear your browser cache and you will be able to see your brand new favicon.