This applet shows number of particles in SHM. All the particles have same amplitude, same frequency but are at different positions at any given moment. The term which would deal with this difference effectively is phase. Each of the particles in SHM can be associated with a particle on the circle and phase is just the angular position of the particle on the circle which represents the given SHM. Phase would increase continuously with time going up by 2PI radians for every oscillation. Since all the particles shown in the applet have same frequency, the phase increases at the same rate for all of them and so the phase difference between them does not change with time.
Phase is one variable which can be related to all other variables easily (position, velocity, and acceleration). That would mean knowing the constants of SHM and the phase would be enough to completely define any SHM.
In adding two or more simple harmonic motions, phase would be a very important term. Imagine two shms of equal amplitude and frequency. If phases were to differ by PI radians the oscillations would destroy themselves and if the phase difference is 0 or 2 PI they would add up. Just visualize these cases. If phases were to differ by PI, the displacements, velocities and accelerations of the two particles would be equal and opposite at any given moment and two such oscillations would add up to zero.