If you don't know what power shifting is, it's a way of shifting used with manual transmissions to reduce the time that it takes to shift from gear to gear. Unlike a normal gear change, in a power shift the driver doesn't let off the gas pedal (unlike speed-shifting, where the throttle is let off very quickly, while simultaneously depressing the clutch pedal and shifting into the next gear rapidly, then getting back into the gas).
With power shifting the clutch pedal is briefly depressed while the shift lever is rapidly shifted into the next gear, keeping the engine in its power band. Keeping the engine in its power band allows it to put down power quicker when the clutch is reengage and the power returns back to the transmission. This is especially helpful with cars that have supercharged and turbocharged engines. It helps maintain full boost between shifts. In many cases, there is a method of cutting the ignition and/or fuel delivery, in a similar fashion to how a rev-limiter works, which prevents the engine from over-revving when the load from the transmission is removed. Many aftermarket engine management systems provide this functionality as either a standard feature or as an option, usually combined with launch control. Power shifting is very dangerous when not done right because you can burn through your clutch, blow up gears and other transmission parts or even possibly blow up the engine. It really does take alot of practice and isn't recommended for everyone to do. Some transmissions do better with it than others.
So with that being said, check out the video below produced by Fatal Motorsports that explains power shifting.
How To Power-shift like a Boss Video