You don't always have to have a tripod with a panorama head to shoot good panoramas. With today's stitching software all you need to do is follow some simple basics to get good usable panoramas straight from the camaera with no tripod.
Step 1: The Parallax Problem
The main reason for using specialized equipment such as a tripo fitted with a panoramic head is to eliminate parallax. So before describing how to shoot panoramas handheld, I think its best to give a brief (non technical) description of Parallax the main culprit for panorama failures. Parallax is the affect that you can see when you hold your finger in front of your eyes and look through it at a distant object, then alternately close one eye and then the other. Your finger moves in relation to the background. what is really happening is that your eyes are looking at the scene from a different viewpoint and introducing Parallax. The same thing happens if we take a photograph of an a foreground object that is in front of something else in the distance (say a car nearby parked in front of a building in the distance). If we took one photo then took a step to the right and took a second image, we would see that the car is in front of a different part of the building. This is bad for shooting panoramas as we need to stitch multiple adjacent photos together to make the entire image. If each photo is different due to parallax then the stitching software will have a hard time.
In the diagram above the addition of parallax will make to adjacent images in a round of panoramas different from each other. This is particularly true when there is an object in the foreground that interacts with a background object. With a lot of parallax you may end up with stitching issues.