Why is group velocity the important velocity? It seems that universally in all systems, group velocity is the relevant velocity but why? I read that this is the speed of information transmission but why is that the case? How do you define where the information is? Even in quantum systems, it seems the group velocity describes the velocity of the particle but why? What makes the group velocity and its definition so special?
 A: I know (radio frequency) waveguides, so I'll answer for waveguides.  I suspect that you can extrapolate from there with decent accuracy.
In a waveguide, you grind through all the math, and you find that the actual phase velocity in the guide is faster than light.  "Woo hoo!" you say, "that Einstein guy was off base!".
But if you take a waveguide and apply a pulse of RF to it, that packet of RF energy travels at the group velocity*.  And in a waveguide, that group velocity is actually slower than the speed of light in a vacuum.
It turns out that any intelligence that you want to communicate via that waveguide has to be carried by variations in the RF energy -- so the speed of information in the waveguide is restricted to the group velocity.  The phase velocity ends up being little more** than a mathematical curiosity.
* well, actually it probably gets spread out because of different group velocities at different frequencies, but bear with me, please.
** I was going to say "nothing more" but I think it bears on how you'd design things like filtering sections and probes to couple energy into or out of the thing.
A: When laymen say they can move a laser spot across the surface moon faster than light with just a flick of the wrist, in violation of special relativity, physicists point out that nothing real is moving across the surface of the moon. The apparent position some energy is moving, but not the energy itself.
Perhaps a better example is the intersection point of 2 nearly parallel lines, like in a Moiré pattern. You can shift the 2 lines and have the intersection point move faster than light; however, the motion is "apparent", in that nothing physical is moving.
That really is a lot like the phase of wave. All it is where a wave is in its cycle, locally. The point at which a wave crosses zero ($\phi=0$) exactly like the intersection point in the Moiré pattern, nothing physical is moving, it's just an apparent location that can move outside the light cone, if the conditions are right.
