BMP (Bitmap) is currently a well known extension, commonly used by Microsoft® Windows' Paint accessory. The Paint program is a very basic editor, and is thus very limited in functions compared some others listed below (last paragraph). Compression isn't the best. Quality is on the weaker side.
JPEG or JPG is a well compressed format of images. This is one of the reasons it is good for the web and most inkjet/bubblejet printers. JPG's are normally multicolored and may contain shades, gradients or blends of colors.
GIF images are also great for the web. GIF's normally contain simple colors and/or shapes like most cartoons and drawings. With GIF images you can also have transparent images and GIF Animations, which are slides of images that can be played on a time sequence. If a small variation of colors are used, compression is great.
TIFF is supposedly the highest quality & low compression, used mostly by the print/press shops. This is not recommended for the web.
We mostly create and convert from PNG (via Macromedia® Fireworks™ software)
and AI (Adobe® Illustrator™). We can store layers and layers of images and other data for compression into JPG, GIF and many others. PNG, AI, PSD (Adobe® PhotoShop Document), QXD (QuarkXpress™ Document) and a few others are well known vector-based graphic designing/editing tools made for small to massive projects.