For IE, Microsoft recommends 16x16, 32x32 and 48x48 packed in the favicon.ico file.
For iOS, Apple recommends specific file names and resolutions, at most 180x180 for latest devices running iOS 8.
Android Chrome primarily uses a manifest and also relies on the Apple touch icon.
IE 10 on Windows 8.0 requires PNG pictures and a background color and IE 11 on Windows 8.1 and 10 accepts several PNG pictures declared in a dedicated XML file called browserconfig.xml.
Safari for Mac OS X El Capitan introduces an SVG icon for pinned tabs.
Some other platforms look for PNG files with various resolutions, like the 96x96 picture for Google TV or the 228x228 picture for Opera Coast.
Look at this favicon pictures list for a complete reference.
This favicon generator can generate all these files at once, with the corresponding HTML code. The generator can also be implemented as a WordPress plugin