All simple harmonic motion systems have these two features:

No loss of mechanical energy.

A restoring force or torque that is proportional to the displacement from equilibrium. The force or torque is also opposite to the displacement - that's what "restoring" implies.

Implications of these facts

The motion of the system is described by an equation of the form
x(t) = A cos(ωt + δ).

The relationship between the acceleration and the displacement is:

a = -ω² x

where ω is the angular frequency of the system.

The period of oscillation is

T

=

2π ω

Consider an object experiencing uniform circular motion. What are the equations giving x and y as a function of time for such an object?

If the object starts at x = A and y = 0, and has an initial velocity in the +y direction, then the equations giving the x and y coordinates of its position as a function of time are:

x(t) = R cos(θ)

y(t) = R sin(θ)

In uniform circular motion the angular velocity is constant, so:

θ = ωt

The equations for x and y become:

x(t) = R cos(ωt)

y(t) = R sin(ωt)

The equation for x(t) for our object experiencing uniform circular motion looks the same as the one we derived for the mass on the spring experiencing simple harmonic motion.

Clearly there's a parallel here - when A = R and the two ω's are equal, the two motions in the x-direction are identical (given appropriate initial conditions).

A light shining in the +y direction would cast a shadow of the circular motion object - that shadow would exactly match the motion of the object undergoing simple harmonic motion.

What happens if, at t = 0, the objects are not at their extreme +x position? All we need to do is modify our equation, shifting it by the appropriate phase angle, δ. The equation for simple harmonic motion becomes:

x(t) = A cos(ωt + δ)

Instead of plugging in θ = ωt we're using θ = ωt + θ_o, and we're calling θ_o δ instead.

If a mass is oscillating vertically at the end of a spring, you might think that we'd need to build in gravitational potential energy to handle the conservation of energy properly. You can do this if you want, measuring the elastic potential energy from the equilibrium length of the spring, but you don't have to.

If you measure the displacements of the spring about the equilibrium position with the mass attached to the spring, this is the point where the spring force cancels the force of gravity. Measured from this point, you just have to consider the kinetic energy and the elastic potential energy - don't worry about gravity.