The greatest advantage to power meters versus older training methods (ie. heart rate) is that there is no lag between your effort and the power shown. Also, power output is independent of rider fatigue, and weather conditions, where as these variables effect the riders heart rate.
The benefits are well explained in this short video:
There are three main types of bike based power meters on the market today.
1) Rear wheel hub by Powertap
2) Crank based power meter by Quarq
3) Pedal based power meter by Garmin
All three of these power meters will pair to most wireless enabled bike computers. The data measured by these meters is then displayed for the rider to view while riding. Having power data while riding allows the rider to accurately pace themselves and make sure they are riding at a sustainable level. As the prices of power meters come down I expect to see them more and more as riders realise the importance of training with power.