How to shoot photos for a panorama?

There are numerous tips and suggestions on how to make better photos for stitching a panorama available on the internet (you can find several links at the end of this page). These are the main, simple but very important guidelines for taking panoramic photographs.

Select the right position
Make sure you stand in one spot and as you turn to take each picture keep the camera close to your body. Stay still and steady, do not lean forwards or backwards while shooting, do not move from the spot you are standing on between shots.
Overlap adjacent photos
Each photo should overlap the previous one somewhere between 30% and 50% (the more overlap the easier the stitching will be), and the last picture should overlap the first one (in case you want to get a full the round pan). If you camera has a special "panorama mode" then always enable it to simplify shooting.
Keep the level right
Use a tripod to line up your images. If you are taking handheld photos, be careful to maintain camera angle. Small deviations are possible and in fact they always will be present if you shoot handheld.
Do not change focus/zoom settings between shots to prevent unneeded distortions in the result or even failure to stitch a resulting panorama. "Panorama mode" in your camera can help you with this.
It is better not to change an exposure for each new shot, however there are not so many scenes in which everything is well lit. If your camera allows you to set exposure manually, please do so and try not to change it dramatically between frames.
Shoot twice
Subject says it all. Shoot twice or even three times, especially if you made a 1000 miles to reach the scene you are shooting.

Links to some other guides:

Digital Focus: Shooting Photos for Panoramas, Part 1 - PCWorld
Digital Focus: Shooting Photos for Panoramas, Part 2 - PCWorld
How to shoot pictures for a 360-degree panorama - CNET Reviews
Panoguide - How to... - Panoguide
Panoramic Photography by Brad Templeton