A favicon is the little icon you see next to the web address for a site in your browser bar. Back in 1999 when they were introduced, by Internet Explorer 5, favicons were a mere 16 x 16 pixel picture that had to be in the .ico format.
Today, the big five (Firefox, Chrome, Safari, Opera, and IE) all support favicons, and the files can be .ico, png, gif, or even .jpg (except in IE). The sizes are also much more varied, depending on the way you view the internet (desctop, mobile, tablet). It seems like each series of devices is looking for a different sized file. A simple solution to adding multiple favicon sizes to your page is The Epic Favicon Generator by Unbox. Upload a file and this generator converts it into any numberer of different sizes depending on what you need. This favicon generator also writes the code that you put in your html header to display the images. All you need to do is copy and paste the code and upload the images it creates to your server.
The Epic Favicon Generator offers 20+ favicon sizes and is free! Check it out!