Let's lok at the potential energy for amass on an ideal spring, as well as an object sliding down a hill.
 
 

Now, look at the potential energy stored in the system of two atoms.  One is taken to be fixed at the origin and the other oscillates about it.

Notice how we can approximate the valley of this complicated curve by a parabola.  This is what we try to do with other complicated systems.  Often if we only have small oscillations about an equilibrium point, we are able to describe that portion of the potential energy curve by a parabola.  We know the solution to this problem; therefore we approximate the complicated system with the simpler simple harmonic oscillator.
 
 

We will only have time to look at a few oscillators this semester.  Some of what we probably won't be able to study includes driven (or forced) oscillators as well as damped oscilaltors