PNG is the best quality for images, JPEG is a close second. PNG are just a lot heavier files so to save space JPEG's are more used.
Generally JPEG y PNG but if you have a image without high quality that doesn't need many colors you could use GIF.
Since there's rarely a case when I don't need transparency in my designs, I usually go with PNG, compressed with a PNG utility for lowest possible file size.
gif for smaller pictures, jpeg or png for larger. png will usually have better quality, but might reduce load time
Only photographs should be in JPG, nothing else. Use PNG for everything, keep a SVG/EPS version on hand to re-make.
I second drhoward - it depends! 1) on the image content (mainly size & colours) 2) on your needs (quality vs. loading time) I use gif, jpeg and png. When in doubt I use the preview and choose the lowest file size that gives me a quality basically indiscernible from the original.
For WWW images I always use a good quality GIF imageset for good quality and good file size. While PNG may be a little better in quality, the file size is too large and can reduce load times, which is very bad. I often use PNG for singular graphics though, such as banners, logos, etc.
IMHO PNG is good for website image. But if you want to put your photos for portfolio site (if you're photographer) try using JPEG.
Depends on what the image IS, in terms of it's color depth and use of color. 24 bit .png is fat, bloated and results in monstrous file sizes - rarely is it a suitable choice for web work despite many people using it for such... It's why I won't use alpha transparency in my images. In VERY rare cases 24 bit .png on low-color count images can be smaller than 8 bit, which is the ONLY time I'll use them. 24 bit .png with alpha is responsible for more fat bloated broken websites than Dreamweaver or Frontpage... Which is like saying it's a worse automobile than a 1984 Yugo GV. 8 bit .png is optimal for images where palette transparency is desired and you want more than 16 colors - it most always provides a more optimized file size than .gif 4 bit .gif is great when you only need 16 colors or less - it's what GIF was originally designed to do and in those cases most always makes a smaller image than .jpg is ideal for photographs. Sure it's lossy, but on a photograph that loss is barely noticeable... Detail loss due to file size restrictions is a FACT of good web design - anyone tells you otherwise is telling you to shoot yourself in the foot so far as traffic to your site is concerned. The pixel perfect art ***'s might bitch and moan about a bit of artifacting and dithering - *** 'em. Generally in those cases they don't know enough about web development to code their way out of a piss soaked paper bag, which is why they usually end up with 2 megabyte pages consisting of several hundred separate files that take a minute to download on broadband. There's a reason my upper acceptable limit for a page on a site is 140k, and I get twitchy at anything over 70k.