Harmonic motion occurs when a system of some kind vibrates about an equilibrium point. The main condition for harmonic motion is the presence of a restoring force that serves to return the system to its equilibrium point. However, due to the inertia of the masses present in the system, they overshoot equilibrium, and if there is no energy lost the system will oscillate indefinitely.
Countless examples of harmonic oscillators can be found everywhere, from the swinging of a pendulum, where gravity is the restoring force, to a mass bobbing on a spring, where Hooke's Law serves as the restoring force, an inductor-capacitor circuit in electronics, atoms in a crystal lattice, and so forth.