Potential energy is technically stored within matter, though a force must be applied to an object in order for it to store potential energy. However, while the energy itself is stored in the mass of the object, another force (gravitational or elastic) must be present to release the potential energy.
Kinetic energy is the energy an object has because of its motion. If we want to accelerate an object, then we must apply a force. Applying a force requires us to do work. After work has been done, energy has been transferred to the object, and the object will be moving with a new constant speed.