How does an airplane move itself in the runway? Basically do they move themselves using their powerful jet/propeller engines to propel themselves forward or do they just use an engine that powers the wheels used for landing? 

I'm not really asking about planes that are readying for takeoffs just the ones that move on the ground like a car would.
The power to move an aircraft from point A to point B on an airport is provided by the same engines that power it in flight unless you hook up a tug, in which case the tug provides the power. There have been experimental developments that provide power to the main landing gear wheels, but there have never been any real implementations of these systems. Small single-engine aircraft can be moved by hand.