You are getting this error in your logs because you do not have this icon file in your root folder.
Some browsers won't call that file or they will cache it in the browser so it won't be requested again. IE is a browser that will cache the icon file on the first request.
As Dormilich said, it's the little icon at that you see next to the name of your website when you open a website in a browser. For example on Bytes, you will see a purple icon at the top next to the name of the page on the tab. This is our favicon.ico.
These images are also used by bookmarking services and when you bookmark a pages. It's an easy way to identify that youv'e landed on a page of a certain website.
You should always create a unique favicon.ico for your website, so your website visitors know that they've landed on a page of your website.