Here is a listing of assorted image file format types with brief descriptions of each.
All but the Photoshop file type are web compatible.

(There are other types of files that graphics programs create and use, but they are not web friendly and therefore are not listed here.)

PHOTOSHOP

Photoshop documents that contain separate Layers are automatically saved in the Photoshop format. The image must be flattened before saving it into other file formats. Graphics programs require that the Photoshop format file be changed to pict, tiff, jpeg, etc. before accessing file.
Files in PHOTOSHOP format may end with a .psd extension.

PNG

Short for Portable Network Graphics, and pronounced "ping."
This new graphics format is similar to GIF. In fact, PNG was approved as a standard by the World Wide Web consortium to replace GIF because GIF uses a patented data compression algorithm. In contrast, PNG is completely patent- and license-free.
The most recent versions of Netscape Navigator and Microsoft Internet Explorer now support PNG.
Files in PNG format end with a .png extension.

JPEG

Short for Joint Photographic Experts Group, and pronounced "jay-peg."
Jpeg allows for substantial compression of file size. This can be as high as 20:1 compression. In addition using JPEG allows the storage of pictures at a much higher color resolution. This allows for greater accuracy and better looking images.
One of the problems with JPEG is that it is a "lossy" compression format. If you begin with a raw image such as a TIFF or BMP, all of the digital information exists. However, when you compress the information using JPEG, some of the information is lost. When the picture is reconstructed, the decompression approximates what the original picture contained. Usually this can be quite good. However, sometimes pictures experience "artifacting" where sharp edges become jagged and smooth textures start looking ragged. The best way around this is to increase the compression quality when you convert the original image. However the tradeoff here is that the amount of space required increases.
Files in JPEG format end with a .jpg extension.

GIF

Pronounced "jiff" or "giff" (hard g), stands for -graphics interchange format- .
GIFs are very internet compatible because of their small size and special features. They are the most common format for buttons, icons, and horizontal rules in html documents. Additionally, the interlacing features allow a document to be downloaded and have the image displayed progressively.
When the GIF format was first invented, 256 colors was a lot. Nowadays, 256 colors is a limitation. Thus for photographs and high resolution images, GIFS don't cut it unless there are less than 256 colors in the image.
Additionally, the nature of GIFs is such that simultaneous display of more than a couple of images can eat up all the colors on your screen leaving you with poorly colored images. Thus for hi-res images with lots of colors, another format such as JPEG or PNG might be a better bet.
Files in GIF format end with a .gif extension.

Animated GIF

A type of GIF image that can be animated by combining several images into a single GIF file. Applications that support the animated GIF standard, GIF89A, cycle through each image. GIF animation doesn't give the same level of control and flexibility as other animation formats but it has become extremely popular because it is supported by nearly all Web browsers. In addition, animated GIF files tend to be quite a bit smaller that other animation files, such as Java applets.
Animated GIF files end with a .gif extension.

Initially, these definitions were found in the Webopedia. I rewrote portions, or otherwise edited what I had discovered. Nevertheless, I want to give credit where credit is due.