What is the power curve
It's an expression of power required to maintain level flight throughout the speed range of the airplane and forms a U shape.  The lowest point of the curve, bottom of the U, is the point of minimum power required for level flight and is normally just below maximum L/D speed (flattest glide speed).  To go faster moves up the U to the right and requires more power.  To go slower moves up the U to the left and also requires more power.

The main bogey man that pilots have to be aware of is the issue of being on the "back side". If you slow to below min power airspeed, slowing any more will require more power to keep from descending; that's the back side.

This becomes a crisis situation in marginal conditions, especially when flying at very high altitudes, and is one of the things that makes flight at the upper flight levels riskier than at lower altitudes; if you are flying at Min Drag  speed or close to it at service ceiling, say FL410, and fly though some down flowing mountain wave air that is sinking at 100 fpm (nice smooth air that you likely won't notice), the autopilot in ALT hold mode will just pitch you up to keep from descending and the jet will slow down. 

If you aren't paying attention and let the speed bleed off too much, you will find yourself in a pickle entering the back side of the power curve, with little reserve thrust to speak of, and will have no option but to tell ATC you have to descend NOW and start down until you can recover your speed.