Learning how to fall is a valuable skill applicable to many different activities. The keys to falling are protecting the important parts of the body, reducing the impact of collision, and remaining flexible. To do this, it is necessary to learn to control the body while falling.
Note: There are as many different types of fall as there are body parts to hurt. This writeup will deal primarily with low level falls, such as from a standing position.
The important parts of the body to be protected are the head, chest, and groin. Protecting these areas is fairly simple. Basically, the groin and chest will protect themselves in most situations, because the arms will protect the chest instinctually and the groin is protected by the legs in most situations. This leaves the head. To protect the head when falling backwards, always try to take falls as low on the back as possible. Keep the head pulled down towards the chest and make sure to keep the tongue out from between the teeth. When falling forward, the best way to protect the head is to never let it make contact with the ground. Instead, try to take the impact on the hands or shoulder.
When falling, always try to reduce the impact of the fall. By this, I mean try to hit the ground as softly as possible. When off balance and falling, the idea is to get as close to the ground as possible before losing balance completely. Reducing the impact also includes controlling which part of the body makes contact with the ground first. Ideally, this should be something soft and/or not important, like hands, shoulder, or butt. If you can spread the force of the fall over a larger surface area, the overall impact will be reduced. This is most important when falling backwards, but the same principle can be applied in any direction.
Remaining flexible is the key to avoiding permanent damage from a fall. Remember the old fable, about the oak and the reeds? The same principle applies here. The easiest way to ensure flexibility is to never ever lock out a joint during a fall. This is the quickest way to break a bone or permanently damage a joint. The knees and elbows should always be allowed to bend in the case of a fall to prevent damage to the rest of the body. Wrists and shoulders don't play as large a part in falling, but they too should remain flexible. However, flexibility doesn't just extend to joints. Tightened muscles can take more damage than loose ones from a fall. Try to avoid tightening a muscle just before impact.
So, to be able to walk away from a fall, remember to stay flexible, protect the important parts of the body, and reduce the impact of the fall. Stay safe and have fun.
My own clumsiness and taekwondo training