I've wondered myself how widespread support for PNGs is in IE Win, but judging by the number of webcomics that use it, I assume it's safe to use these days.
As for the original question, you can think of PNG as basically the same as what GIF covers, with the potential for extra features. If you're compressing a photo-type graphic, JPEG is the way to go, because that's designed for photos. If you're compressing lineart or a graphic graphic (bars, large areas of flat colors, etc), then either PNG or GIF will work, but PNG gives you the added option of more than 256 colors, and it's an open format.
GIFs, unless you're going for compatibility with very old browsers, are best limited to animations and maybe things with transparency these days (though PNG actually has transparency features far surpassing anything a GIF can do, they might not work on some recent versions of IE).