in order to execute simple harmonic motion ,a systum must have

Requirement of mean position :It means there should be a mean position available in the system. It is the position where net force on the body is zero.
Restoring force/ restoring torque: If we observe the definition the motion should be to and fro. For to and fro motion we need a force that always bring the particle towards mean position. Such a force is called restoring force(and restoring torque in case of angular oscillations).For example in case of simple pendulum gravitational force acts as restoring force. In case of oscillation of a spring block system spring force acts as a restoring force.
Variable force : Only a variable force can result in oscillations. A constant force can never result in oscillations. For example spring block system oscillates and in this case spring force provides the restoring force which is a variable force.
Stable equilibrium position: Point no. 1 says we need a mean position. The another name of mean position is equilibrium position.There are three types of equilibrium namely stable, unstable and neutral equilibrium(although we have a meta stable equilibrium as well).Only a position of stable equilibrium position(mean position ) would result in oscillations.
External disturbance: When the particle is initially in mean position the net force acting on it is zero. So by itself it will not start oscillating. Some external disturbance is initially required for it to start oscillating