What is the significance of dispersion relation i.e. frequency vs. wave vector relation? There are lots of pages in my textbook dedicated  to this sole relation and frequency vs. wave vector graph. I do not understand what are practical significance and uses of the relation and the graph. Can you please point out their practical significance and uses in real world?
 A: Take into account the simple case of a wave that satisfy the D'Almbert equation (for simplicity 1-D case): $$ u_{tt} - c_{0}^2 u_{xx} $$ You know that the most general solution of this equation is given by the superposition of two profiles that does not change in form during the motion:
$f(x-ct)$  that traslates towards positive $x$ and $f(x+ct)$ that traslates towards negative $x$. 
The general solution reads: 
$$ u(x,t) = f(x-c_{0}t) + f(x+c_{0}t) $$
It is the so-called "stationary wave", that is a superposition of two profiles with the same frequency, that does not change during the evolution according to the D'Alambert equation. 
Now let's get to the point: the D'Alambert equation above is a linear (superposition of solution is a solution yet) and NON DISPERSIVE equation, the fact that is not dispersive, practically means that you can change the frequency (or the wave-number) of your traslating profiles, without changing the phase velocity of propagation, that in any case remains 
$$c= \frac{\omega}{k}=\pm c_{0}$$.
You can see that in this first case the dependece $\omega(k)$ is linear and
as a consequence of this fact, you will have that your traslating profile will not change in time, as written above.
This also can be thought in terms of Fourier components: just think about that in this way you can describe a stationary wave like a sum over components of different wave-number. In other words, the most general way to write the solution can be by developing a Fourier series, incuding all the possibile wave- vectors $k$ in your solution, without losing of generality.
For example, a non dispersive wave is the electromagnetic one in the vacuum.
On the other hand, a DISPERSIVE wave-equation, is an equation in which the dispersion relation is something like $$c= \frac{\omega(k)}{k}$$. Where the dependece $\omega(k)$ is not linear in this second case.
This cleary imply that the phase velocity of your wave will change, if you modify the  frequency (beacause $\omega$ depends on $k$) and so, in this case you cannot have that waves with different frequencies (or wave-numbers) propagate with the same phase velocity. As examples of dispersive wave, think about surface waves on deep water, or electromagnetic waves in a medium. 
A: Dispersion is of practical significance as a measure which characterizes transparent materials (or regions in which waves propagate); as the possible characteristic dependence of the refractive index $n$ of the material or region under consideration on the wave frequency ($n[~\nu~]$); or closely related: expressed as  the possible characteristic dependence of refractive index on wave length ($n[~\lambda~]$).
Even more practically, it can be desirable to minimize the dispersion of an optical system, for instance, by suitably combining optical elements (lenses) of appropriate shape made of different materials (glasses) with given different dispersion characteristics into one achromatic lense.
A: Lovely dispersion relation! Whenever you have a wave-like motion of any kind, the first thing to do is find out how $\omega$ relates to $k$. If there is a square root of minus one ($i$) involved, then you learn that your waves are decaying as they go. If the relation is simply proportional, then you have simplest case, and waveforms retain their shape as they go. If you have $\omega \propto k^2$ then it is like de Broglie waves (in non-relativistic case) and the proportionality constant tells you the effective mass. If you have $\omega^2$ related to $a^2 - k^2$ then it is like a plasma, and you get either transmission or reflection, depending on $\omega$. Finally, in all cases the equation allows you to find how both group and phase velocity relate to frequency, and hence also to each other. All from this one equation! Lots of lovely physics!
