A constant speed propeller is a feature of propeller-powered airplanes, both piston engined and turbo-prop. In the simplest form of airplane, the propeller is fixed to the crankshaft of the engine (maybe not directly, but at the least it is directly driven). As a result, changing the power settings of the engine will cause it to change RPMs, which will make the propeller spin faster or slower - and this is how you change the power settings on your aircraft.
This isn't the most efficient way to do things, however. Aerodynamics means that for a given blade shape, the propeller will be most efficient at a particular speed. Also, the engine will produce different levels of power at different RPM values - just like in your car, where it will produce the most torque at a particular range of RPMs.
A constant speed prop is a system whereby the propeller's angle of attack can be modified dynamically. In simpler terms, the blades can 'twist' so as to present more or less of their surface area to the air around them. Generally, this is done using a hydraulic system that lets the pilot manipulate the desired RPM range. When the highest power is desired, and efficiency isn't as much of a problem, the prop is set to 'high RPM' - for example, on takeoff. Once the aircraft has reached an altitude where the high power setting isn't required, the propeller can be set to a lower RPM - higher 'bite' - for efficiency and to avoid straining the engine's moving parts. Think of it like gears on a bike or car - when starting out you want low gear, which results in the fastest and easiest rotation of the engine or pedals. At high speed, when not as much torque is necessary to maintain speed, it's much more efficient to shift up so that a slower rotation can be used.
The downside to this system is increased complexity in both the airplane's systems and normal procedures - the pilot has to manage an additional factor to change power settings. The additional systems required will also exact a weight penalty.