favicon.ico is the little icon of the website you are visiting that appears next to their URL in the address bar of web browser. For instance, this website display a small icon symbol of a letter C with some blue light rays radiating from the letter next to the URL.
If you are using a modern-day web browser (such as Firefox, Opera, Safari, etc), you will see it. Older browser, such as Internet Explorer, does not use or show that small icon next to the URL.
If you are creating your own website, all you need is to put the “favicon.ico” image file in the home directory, and the browser will automatically locate it and display it. If the “favicon.ico” file is not found, the browser will display a generic icon instead.