Here are the major formats and how to use them.

.jpg - lossy compression, but high quality when it comes to photos and and art with a lot of colors.

.gif - restricted to 256 colors, but good for things with lotsa of solid colors (like logos, icons, etc.); also allows for transparency

.png - lossless; only use if perfect fidelity is crucial. otherwise, go with .jpg

Using .gif files in web pages is generally frowned upon these days. The patent for the Graphics Interchange Format (gif) is owned by Unisys, who went around a few years back and sued a bunch of websites who were using gifs without paying royalties.

Legal issues aside, Portable Network Graphics (png) compression will always give you a smaller file for the same color depth as gif. There is a really nice comparison between jpg, png, and gif and when you should use each here: http://www.r1ch.net/img-formats/