The motion of a body is simple harmonic if the acceleration is directed towards a fixed point and proportional to its distance from the point.
In other words, if a body moves back and forth along a straight line or rotates back and forth around a fixed point, and its acceleration is always directed towards a fixed point and proportional to its distance from that point, then the motion of the body is simple harmonic.
A simple example of this is the motion of a mass attached to a spring. When the mass is displaced from its equilibrium position and released, it oscillates back and forth, with the acceleration directed towards the equilibrium point and proportional to the distance from that point. This motion is simple harmonic.
Another example is the motion of a pendulum. When a pendulum is displaced from its rest position and released, it swings back and forth, with the acceleration directed towards the equilibrium point and proportional to the distance from that point. This motion is also simple harmonic.
In summary, the motion of a body is simple harmonic if it oscillates or rotates back and forth around a fixed point, and its acceleration is directed towards that point and proportional to the distance from it.