Power is similar to strength, however speed is now also a consideration. Power comes in handy in sports where a quick force is required for a limited time, but usually more than just one repetition. for example, sprinters require a lot of power. They need the strength to move them, but they must also move quickly.
Power is harder to train than just strength. It requires specific training pending on the activity performed. Ideally, an athlete should first strength train and reach adequate strength levels. Then, gradually switch the training to sport specific activities to develop the power movements needed.
Power training usually does not produce large muscles. But in some athletes power training is good for developing tight muscles. In combination with low body fat levels this will have a pleasing look.
Most people who are just interested in general fitness don't need to worry too much about power. But by adding some power training in your workout can provide something different than your usual training routine.
Measuring power must involve speed and strength. A very simple test is to find a hill that you can run up. Pick a specific distance then run up the hill as fast as you can. The hill provides a greater resistance, compared to level ground, which requires more strength. While running as fast as you can is a measure of power. You can also do a similar test on a bicycle.
Of course you can also use a weight machine, or dumbbells, for this test as well. Pick the resistance (weight) and see how many repetitions you can do in a pre-specified time frame, e.g. in 30 seconds. The time frame used shouldn't be too long because then muscular endurance will come into play as opposed to just power. If the muscle begins to burn then the time used is too long.