Power is the ultimate combination of the two most fundamental human factors of survival: speed and strength. We can hear your brain now: So what? What will being more powerful do for me? Will it help me look better in the mirror?
Here's your answer: The advantage of power training is that if you improve your rate of force development, you inevitably improve neural recruitment, which means you'll activate muscle fibers more efficiently and effectively. In the long run, this means that when you do pure strength or hypertrophy-type training, you'll activate more fibers and increase muscular size. Increasing power is also great for busting through training plateaus, a problem that every trainee is bound to face eventually.
The Time Principles
Reducing the time it takes to lift a weight is the rate part of the power equation. The speed principle advises you to move as fast as possible; the next step is taking a near maximal weight and moving it quickly. A strength rep, speed rep and power rep will all have a different application of time and force because of how much weight is being used. Whereas strength is best produced with loads of over 85% of your 1RM, and speed with 30% or less, power is best produced with somewhere between 30%-60% of 1RM.
Another way to determine appropriate resistance is to try completing sets within a specified time period. For example, use about 60% of what you would normal use for an eight-rep set (which is considerably lighter than 60% of 1RM) and try to perform eight reps in 15 seconds or less. If you can work up to using 80% of your normal rep weight in 15 seconds, you're truly maximizing power and it's time to increase the amount of weight you're using.