For images on the web, each format has its pros and cons. For photograph-type images (ie lots and lots of colours, no hard edges) use a JPEG.
For icons and the like, you have a choice between PNG and GIF. GIFs are limited to 256 colours. PNGs can be formatted like GIFs (ie 256 colours, with 1-bit transparency that will work in IE6), but for small images they're slightly larger than GIFs. 24-bit PNGs support both a large gamut, and alpha transparency (although it's troublesome in IE6).
PNGS are your only really sensible choice for things like screenshots (ie, both lots of colours and hard edges), and personally, that's what I stick with most of the time, unless I have something for which JPEG is more suitable (like a photo).