David,
Variable pitch props are just that, the pitch is variable. One type of Variable Pitch prop has a constant speed mechanism as well, this is the constant speed type. This is probably the most common, particularly in small to medium Pipers/Cessnas/Maules etc.
There are other types, ground adjustable, two and three position, and just variable from the cockpit.
The advantage of constant speed props is that the automatic system keeps the prop speed contant by automatically varying the pitch. This is typically done by controlling the oil pressure feed into the propeller hub. There is a governor which adjusts this oil pressure (and hence the pitch) to keep the speed to that set by a lever in the cockpit.
The pilot therefore just sets the required engine RPM on a third control (Throttle, mixture and now prop speed ) which is usually blue. Opening and closing the throttle does not change the prop speed. To measure the power, a manifold pressure gauge is used and this indicates how wide the throttle is open.
This automatic speed control mechanism means that the engine is running at the optimum speed at all times, takeoff, climb, cruise and descent.
This gives great get up and go on takeoff and a great climb rate whilst maximising the cruise speed.
One other big advantage is in the descent. With a constant speed prop you just point the nose down and leave everything set. The engine cannot overspeed as the prop governor prevents it by coarsening the pitch. Because you don't throttle back you can usefully convert all your height to speed and there is no shock cooling (as the throttle is still open).
The downside is that there is a higher maintenance cost for CS props.
There are many subtleties with different types so please consider the above a general simplified answer which will apply to the majority of small singles with VP props but there are less common versions which will differ.