EDIT: I stumbled upon this wonderful repo the other day. It renders this entire blogpost obsolete.
The favicon has come along way since its humble 16 × 16 px beginnings. Screen resolutions are skyrocketing and visual representations of websites are needed in different applications on different devices. This post is intended to be a collection of many image sizes that can (should?) be included on modern websites.
standard favicon.ico file Used to be just 16 × 16 px, but nowadays larger dimensions are necessary as well. E.g. pinning a website to the taskbar in Windows requires 32 × 32 px, double that for high pixel density displays and you're at 64 × 64 px. I recommend making separate versions instead of just making the largest one and letting the browser scale it since you want more control at these really small sizes. You can conveniently merge them all into one .ico file using GIMP.
iOS home screen icons These are used as icons when pinning websites to the home screen. There are 4 dimensions to cover 2 devices with 2 screen types each:
144 × 144 px (iPad with Retina display)
You can also specify whether the image is precomposed or not (if not, the device will add some gloss and other effects over the image). More in Safari Web Content Guide.
Opera Speed Dial icon Used as the thumbnail for Opera's Speed Dial screen. The minimum required dimension is 114 × 114 px, anything over 256 × 160 px will be resized, details on Dev.Opera.
Windows 8 Start Screen tile Displayed on all devices with Windows 8 when you pin a website to the Start Screen. The required dimension is 144 × 144 px, Microsoft provided an official tool that helps you with the process.