Often we come across different types of velocities, associated with wave motion, in one way or the other. For instance, we talk about the 1) particle -velocity, 2) wave velocity or phase -velocity and group velocity. Each has got a distinct definition as well as physical meaning for it. Before we proceed to discuss different types of velocities, it is worthwhile to bear in mind that a wave motion is a collective response of a large number of oscillators through which the disturbance goes, one by one,. Also, each of these oscillators undergoes a simple harmonic motion (S.H.M), at its own location. The very important fact here is that these individual oscillators of the medium, through which the wave progresses / advances, do not progress move along with the traveling wave. The motion of these oscillators about their mean positions can be transverse or longitudinal. What we observe as a wave motion is the locus of different particle phases. In other words, the waves as observed by us are the phase relationships of different particle oscillators coupled and not their progressive motion through the medium they constitute. 

1) The Particle velocity  -  It is simply the velocity with which the simple harmonic oscillator moves about its mean or equilibrium position. As already discussed, this motion can be transverse or longitudinal direction of wave motion. 

2) The wave velocity or Phase velocity -  As we all know, crests and troughs represent planes of equal phase. the wave or phase velocity has been defined as the velocity with which planes of constant phase advance through the given medium.

3)  The group velocity-  It is an established fact that waves rarely occur/exist as single monochromatic  components. Instead that , what we have is a superposition of a number of waves of different characteristics. The resulting composite entity is called a wave group.