Simple Harmonic Motion is a specific kind of periodic "back-and-forth" motion. Specifically, the displacement is proportional to a (possibly phase-shifted) sine wave.

Simple Harmonic Motion arises when there is an item to be displaced, and the restorative force is proportional to (but in the opposite direction of) the displaced. The movements of simple pendulum and a mass on a spring are examples of SHM.

SHM can be considered as circular motion, with constant angular velocity, in one dimension. For example, circular motion with constant angular velocity in the complex plane results in SHM along the real axis.

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