This id the very basics, but hopefully the layman will help you visualize.
Potential energy is the type of energy that is stored up and ready to perform work on an object. One common form is gravitational, where an object--like say a bowling ball--is lifted up and held there. Another form of potential is elastic, where something is bent from its original position into a new one, such as a bow pulled back holding an arrow.
Kinetic energy is the release of the potential to do 'work,' or movement energy. In the gravitational potential energy example, the bowling ball is released, and it falls back down to the ground. In the elastic potential energy case, you let go of the bow string, and the bow rights itself. That energy is transferred to the arrow and lets it fly.
Kinetic energy and potential energy is essentially a zero-sum situation, though you must not forget about friction, which converts some of that energy into heat. That's why when you put a ball at the top of a U-shaped ramp that each time it makes a pass to the other side, it doesn't reach quite as high as the starting drop from before: the ball pressing against the ramp as well as the curve of the ramp forcing the ball to change directions induces friction, causing some of that kinetic energy to become lost as heat energy as it rises up the other ramp end (turning that kinetic energy back into potential energy, until it drops back down again to become kinetic energy once more).
This continues until all of the energy is turned into heat and the ball stops at the center (re: bottom) of the ramp.