BMP - Raw format, completely lossless, comes in several different formats (24bit, 256 color, 16 color, monochrome); but since it's raw it's very large.
JPEG - High compression even at high colors, best used with real-life photos as opposed to cartoonish low-color images; often lossy but less noticeable with more detailed pictures.
GIF - Highly compressible, supports transparency and is most often used for clipart and animations; limited to 256 colors.
TIFF - I don't know what tiff is...
PNG - Like a cross between JPG and GIF that also supports transparency; often has different issues among web browsers and thus is looked down upon by internet perfectionists
GIF and JPG are the most used image formats on the internet.