Programmers, on the other hand, are that rare breed of human that just can't leave information well enough alone. They need to know how every byte is arranged to see if someone knows something that they don't (and often snicker contentedly to themselves when they find that it is really they that know more). Programmers will then use this information to write code that may never see the light of distribution, but nevertheless, they will have had fun and gained enlightenment from writing it.

Web Graphic Formats

Graphics Interchange Format (GIF) pronounced "JIFF" like the peanut button*
Compuserve GIF - GIF87a
256 colors or less, 216 colors on the Web
best for aliased line art
8 bit (requires conversion from 24 bit)
lossless compression along horizontal axis
interlacing
dithering positioning pixels to simulate a color not in the palette

Joint Photographic Experts Group (JPEG, JPG, JPE) pronounced "JAY-PEG"
24 bit, 16.7 million colors
lossy compression
longer decompression time than GIF
good for continuous tone images such as photos
but not line art as it introduces blur

Portable Network Graphics (PNG) pronounced "PING"
Unisys claims for royalties for LZW patent
24 bit, 16.7 million colors
masking (transparency)
public domain
slow acceptance

Non-Web Graphic Formats

Windows Bitmap (BMP)
The Windows Bitmap file format is the standard file format used by Microsoft Windows. Bitmap files can contain either 2 (black and white), 16, 256 or 16.7 million colors. Most Windows Bitmap files are not compressed. It is possible to save 16 and 256 color images in a compressed format called Windows Run-Length Encoded (RLE)

Tagged Image File Format (TIF)
The TIFF format was developed by Microsoft and Aldus Corporations as a portable method of storing bitmap images. TIFF files come in monochrome, 16-color, 256-color, 16-color grayscale, 256-color grayscale and true color (24-bit) varieties. TIFF files exist in both LZW compressed and uncompressed formats. The compression format offers a high degree of compression. However, certain older paint programs that use an early version of the TIFF format may have difficulty reading compressed TIFF files. If you find that a program will not read the compressed files, try saving the file in the uncompressed format. The default file extension for TIFF files is “.TIF”. This format type is commonly used in the pre press or print industry.

AOL Only format ART pronounced "Proprietary Garbage"
America Online has developed a proprietary image format that is incorporated into it's browsers. When you're browsing the web using the default AOL settings, it converts the graphics that you see into a format called ".art." This can lead to problems not only with viewing graphics on the web, but also when downloading and sending these converted images to people that don't use AOL. (Even Photoshop won't open an ".art" image.) That image you thought was a J-Peg or GIF, when downloaded while using AOL is really an "ART" even though the filename extension might read jpg or gif.

The workaround for this is very simple. Go to your AOL preferences, then to the "WWW" tab, and look for the checkbox that says, "Use compressed images." Make sure to UNCHECK that box. You may have to close your browser, and then relaunch it for the changes to take effect.

Then write a nasty letter to AOL letting them know that you don't appreciate paying their ridiculous rates for dial up and still having to put up with inferior proprietary image compression which wiser non-AOL users can't and don't wish to see.

Optimize (which to use and how to compress it)

I tend to use 10% J-Pegs on most photo-type images here on my Websites, as they give me the best results, load time verses quality. I use GIF (pronounced Jiff) images for clip-art or other images with few colors like text graphics.

Examples
Below are some examples of GIFs, J-Pegs and Pings. Compare the size and quality of the photo portion and the text part.