Simple harmonic motion is any motion where a restoring force is applied that is proportional to the displacement and in the opposite direction of that displacement. Or in other words, the more you pull it one way, the more it wants to return to the middle. The classic example of this is a mass on a spring, because the more the mass stretches it, the more it feels a tug back towards the middle. A mass on a spring can be vertical, in which case gravity is involved, or horizontal on a smooth tabletop.