Favicons were introduced in 1999 by Internet Explorer 5 (ref) and standardized by the W3C a few months later. They are a small graphic that represents the website.

What is the primary favicon file?
favicon.ico. And just to make sure: this is not a PNG renamed to favicon.ico. Although some browsers are easygoing enough to forgive this mistake, ICO is a different format, one that supports multiple versions of an image.

What dimensions should favicon.ico be?
favicon.ico is a format originally conceived by Microsoft and the other vendors have gone along with it. Microsoft recommends 16×16, 32×32 and 48×48. Yes, a single ICO file can contain multiple graphics.

What format do we need to support mobile platforms?
You need PNG icons.
And Apple Touch icons.
And Windows 8 tile graphics.
And a file named browserconfig.xml.

Which sizes should the PNG icons be?
A: 96×96 for Google TV
B: 196×196 for Android Chrome
C: 228×228 for Coast by Opera
D: All of the above
Answer: D, and more. Like the 160×160 picture for older versions of Opera Speed Dial (long gone), or the 128×128 icon for Chrome Web Store, here it really depends on the platforms you want to support.