Take a small weight and tie it to the end of a string.  Let the weight hang straight down, then pull it over to one side by a small amount and let it go.  Once released, the weight will swing back and forth, swinging down from one side before passing through the point where the string is perfectly vertical again, then swinging back up until it comes to a stop on the other side.  As the weight swings back and forth, it is undergoing oscillatory motion, moving back and forth about an equilibrium point.

Many systems exhibit this type of behavior.  If you pull a mass attached to the end of a spring and then let it go, the mass will bounce back and forth about an equilibrium point.  If you push an inflated pool toy down by a small amount and then let it go, it will bob up and down in the water about the level where it was initially floating.  At the atomic level, the bonds between atoms within solids often exhibit this same behavior, bouncing around some central position in the lattice.  In each of these cases, the system exhibits this oscillatory behavior because there is a restoring force that attempts to pull the system back to a stable equilibrium point.  The simplest form that oscillatory motion can take is when the magnitude of the restoring force acting on an object is directly proportional to the object’s displacement from equilibrium.  In this case, the restoring force satisfies Hooke’s Law, `F=kx`, and the object exhibits simple harmonic motion.