For full colour photos, JPG are a good choice due to the balance between filesizes and image quality. They are a lossy format i.e. an image saved in JPG format will be slightly different to the original picture due to the way the image compression works; however, it makes up for it with much smaller filesizes and you can adjust the quality/compression ratio.
Gif format are a lossless format but only support 256 colours. On the upside, low colour images are compressed well and they allow on-off transparency. They are also the only reliable format for animated graphics on the web.
PNGs are, generally speaking, a better alternative to GIFs. PNG often give better filesizes than GIFs with a wider colour depth (true-colour as opposed to GIF’s 256 palette). They also allow alpha transparency, that is graduated transparency as opposed to GIF’s on-off transparency (however, IE’s support for PNG transparency is clunky). The only downside to PNGs over GIFs is that they don’t support animation.
In short, if you have a photo or a true colour image, JPGs are usually the best choice as the files are generally smaller. PNGs are the best choice for other uses such as for web images such as logos and backgrounds. If you want animated images, then choose GIFs.
All other formats (TIFFS, BMP etc) are not web-friendly and should not be used.
Hence, the answer to your question is that it depends on the intended use. The 3 formats between them cover all the bases.
