Most Canon Digital Cameras, like my Powershot A700 come with a stitch assist shooting mode:
Canon also supplies stitch assist software (Photostitch) which makes it really simple to combine images to create amazing panoramic pictures as shown here:
The software does a great job lining up your pictures, but if you need to edit/tweak the seams, you can do that too.
A tripod helps but isn't needed to get started. In fact, all wallpapers on this website were taken without a tripod. If I can do it, you can do it!
With 8 megapixel cameras getting cheaper and cheaper, a question arises... why stitch at all? After all, an 8 megapixel image is 3504 pixels wide -- making it large enough to crop a 3360x1050 slice. To me, though, it's all about framing and resolution. 3504 doesn't give you much room to scale down, and a crop of a single 8 megapixel shot makes it hard to frame. In this example, the highlighted area of this picture is 16:5 (the aspect ratio for 1680x1050) while the source picture is roughly 3:2.
Two (or even three) stitched 6 megapixel shots gives you a little more room to work with: approximately 4600 pixels for two and 6200 for three. A 3360x1050 scaled-down and crop of two stitched 6 megapixel shots looks like this:
As you can see, the aspect ratio of the stitched image is closer to the final image which is linked below:
One trick is to place the stitch-line at the center of the image when you crop so that it is split between the monitors. Even though the stitch on the above image isn't very good, it's not noticeable at all when placed across two 1680x1050 monitors.
When consumer cameras are in the 12 megapixel range, one shot will probably do it, but by then, your monitors will probably be at 2400x1500. :)
Ken Rockwell has a bunch of good info about stitching with a Canon camera here. If you know of additional reference sites, comment here, and I'll hook up the links.