The (differential) equation of motion for a particle of mass m on a spring with spring constant k and a displacement x from the equilibrium position is
m d2x/dt2 = -kx
When you solve this, you find a solution of the form
x= A sin(ωt - ∅)
∅ is an arbitrary value for the phase of the oscillation.
That's where the variable ω comes from. Trig functions involve angles so ω has the dimension of an angle divided by time. Hence it's referred to as an angular frequency. There is one other point and that is that ω is in radians (as all good angles are) so is 2∏f, where f is the frequency in cycles per second (Hz).