There are two good reasons: to make your image files smaller, and to store
24-bit-per-pixel color data instead of 8-bit-per-pixel data.

Making image files smaller is a win for transmitting files across networks
and for archiving libraries of images.  Being able to compress a 2 Mbyte
full-color file down to, say, 100 Kbytes makes a big difference in disk
space and transmission time!  And JPEG can easily provide 20:1 compression
of full-color data.  If you are comparing GIF and JPEG, the size ratio is
usually more like 4:1 (see "[4] How well does JPEG compress images?").

Now, it takes longer to decode and view a JPEG image than to view an image
of a simpler format such as GIF.  Thus using JPEG is essentially a
time/space tradeoff: you give up some time in order to store or transmit an
image more cheaply.  But it's worth noting that when network transmission is
involved, the time savings from transferring a shorter file can be greater
than the time needed to decompress the file.

The second fundamental advantage of JPEG is that it stores full color
information: 24 bits/pixel (16 million colors).  GIF, the other image format
widely used on the net, can only store 8 bits/pixel (256 or fewer colors).
GIF is reasonably well matched to inexpensive computer displays --- most
run-of-the-mill PCs can't display more than 256 distinct colors at once.
But full-color hardware is getting cheaper all the time, and JPEG photos
look *much* better than GIFs on such hardware.  Within a couple of years,
GIF will probably seem as obsolete as black-and-white MacPaint format does
today.  Furthermore, JPEG is far more useful than GIF for exchanging images
among people with widely varying display hardware, because it avoids
prejudging how many colors to use (see "[8] What is color quantization?").
Hence JPEG is considerably more appropriate than GIF for use as a Usenet
and World Wide Web standard photo format.

A lot of people are scared off by the term "lossy compression".  But when
it comes to representing real-world scenes, *no* digital image format can
retain all the information that impinges on your eyeball.  By comparison
with the real-world scene, JPEG loses far less information than GIF.
The real disadvantage of lossy compression is that if you repeatedly
compress and decompress an image, you lose a little more quality each time
(see "[10] Does loss accumulate with repeated compression/decompression?").
This is a serious objection for some applications but matters not at all
for many others.