Weight transfer is a very important concept, for motorcycles, cars, as well as bicycles and what-have-you. Most people do not realize the mechanisms and effects of weight transfer and that holds them back from becoming a better rider, driver, skier etc!
Weight distribution is about traction and control. For instance, when driving a car, you have no control when going fast over the top of a crest, since the wheels are unloaded - you can't brake effectively or steer. You must have noticed this feeling of weightlessness in that situation, but probably didn't acknowledge its effect on the wheels, traction and control.
Understanding weight transfer, practicing it until it becomes second nature is critical for both safety and going fast.
Here's why understaning and managing weight transfer is critical: many crashes are due to locking up the front wheels under braking. This, more often than not, is due not to braking too hard but simply not waiting for the weight to transfer forward, increasing the traction!
While braking, the weight transfers forward and the rear wheel may even come up. This is why you should move your body back in preparation for braking, to compensate.
While accelerating on the other hand, the opposite is true: the front may lift and all the weight transfers to the rear. You then move your body forward to compensate.
When you are coming down a steep hill, the weight also naturally is on the front wheel, so you keep your butt back on the seat, to keep the bike from going over.
While going up the hill, the opposite occurs: weight is on the rear so you compensate by moving to the front.
Going over a log or other obstacles starts with hopping the front wheel over and you want to have it unloaded. You can help it by moving back, use some throttle to lift it or at least unweigh it (compress the suspension before the log and then use that to unweigh the wheel over the log).
While cornering, you also want more traction on the front wheel(s) and keep the bike/car on the proper line - otherwise the front will wash out, if you don't have enough weight/traction for the speed. Thus you move your body forward, in sit-down cornering, as extreme as sitting on top of the gas tank!
The thing about weight transfer is that it's not instantaneous: it takes a small amount of time for the front wheel to be loaded when braking for instance and, if you don't give it time to build up more weight/traction, it may lock up.
This is one of the major causes of accidents and wipe-outs: locking the front wheel by squeezing the brakes too quickly and not allowing the time for the weight to shift forward increasing traction.
Braking has its own detailed post.