A flavicon is a small image, usally 16 x 16 pixels, that appears next to your blogs name. The one that blogger uses is white within an orange square. I just did one today. Mine looks like this, using the University of Michigan's yellow and blue colors. Very simple but easy to see.
The problem with doing this is that Internet Explorer, Firefox, and Chrome all want the see the HTML code for referencing the favicon in different places in your blog code, and Blogger tries to hijack this so the Blogger orange favicon is shown and not yours. I was going to provide instructions on how to do this but I realized people will come to me if it does not work- so I have decided not to!
Still, if you see this on your own browser next to my blog title I guess it worked!