Potential energy is energy stored in an object or system of objects. It can be related to its position, the bonds in its chemical structure, its potential for radioactive decay or even its shape, to give a few examples. It has the ability or potential to be transformed into more obvious forms like kinetic energy. Potential energy and kinetic energy are what make up mechanical energy.
Imagine a ball just above the surface of a planet, like Earth. Now imagine lifting the ball off of the ground and holding it in your hand. Energy is required to lift the ball, but after the ball stops moving, the energy used to lift it does not disappear. It has only been converted into gravitational potential energy. Dropping the ball will release the stored energy; as the ball falls, it regains this energy in a kinetic form as it moves. When the ball hits the ground, the kinetic energy will be absorbed by the ground as heat or released as sound that you can hear.
Likewise if a Slinky is stretched and let go, it will pull itself back to its original shape, that means that there's the potential for energy in the Slinky's shape. This is another form of potential energy called elastic potential energy.
Types of potential energy include:
The University of Colorado has graciously allowed us to use the following Phet simulation. Explore this simulation to see how gravitational potential energy and spring potential energy go back and forth and create a changing amount of kinetic energy (hint: click show energy before hanging a mass):