Why are a motorcycle's front brakes more effective than back? I've noticed that when I apply the front brakes on my bike it stops quite quickly. If I apply the back brakes at the same speed, it skids rather than stopping quickly.  Why?
 A: The braking force acts between the tyre and the road. The centre of mass is above this point so there is a rotational effect which increases the force going down through the front tyre and decreases the force going down through the rear tyre. Because the amount of braking force the tyre is able to produce is limited by the amount of force going down through the tyre the front is therefore able to produce more braking force than the rear.
In reality, of course, the suspension geometry is not rigid and the rider also moves forward in their seat so the effect is a bit more complicated than I've described here but the basic point stands. The front brakes are more effective because the force of braking acts below the centre of mass and this produces a rotational effect.
A: Braking acts to stop the front tire. Friction acts at the contact patch under the front wheel to introduce a vector force directed towards the back of the bike. Since the force is not directed through the center of mass of the motorcycle/rider system, it introduces a moment or torque that acts to rotate the motorcycle and rider such that the back tire begins to come off the ground, or reduces the force conducted through the rear tire caused by the weight of the bike. The available braking or frictional force at the rear tire is proportional to the weight force directed through the rear tire.
So as the vehicle is braked, available braking power at the rear tire is reduced with positive feedback.
As the vehicle is braked, available braking power at the front tire is increased with positive feedback, for some given constant acceleration.
An analogy to running is that if you attempt to stop on a dime, you will fall on your face.
A: Using the brakes on the front of the bike causes your weight to shift forward.  Additional weight allows more force before the tire will slip (skid).  If you brake hard enough the back tire of your bike will lift up and at that point all of the mass is distributed on the front tire.
Remember the maximum force is $F_{max} = \mu F_{normal}$ and $F_{normal}$ is proportional to the distribution of weight on the tire so as weight shifts forward $F_{normal}$ increases and therefor the maximum stopping force $F_{max}$ increases.
Braking with your back tire doesn't shift weight onto the back tire so its stopping power doesn't increase.
Wikipedia explains this in great detail in the Bicycle and motorcycle dynamics article.
A: Lots of excellent answers here, but for fun, lets think about this backwards. Imagine you have the worlds first and only FRONT wheel drive motorcycle, and your rev it up and pop the clutch. What kind of launch do you think you would get with very little weight on the front tire? The reverse is true during braking when the deceleration shifts the weight of the bike to the from wheel.
A: Any change in speed is acceleration. Accelerating is equivalent to gravity being different. Braking slows you down, which means you are accelerating backwards, which means you feel gravity pulling you in the same direction as on a downhill slope. As on that downhill slope, more of the your bike's weight is supported by front wheel and less by the back wheel.
