What is the significance of the Debye temperature from a materials perspctive? If I look at a table of different metals and their Debye temperatures, what does the variation in these temperatures tell me about these materials?
 A: Since the question is rather vague, I will just give you some key points:
Debye's model treats oscillation modes of a solid as sound waves (phonons) with frequency $\omega(\mathbf{k})=v|\mathbf{k}|$ ($v$ the sound velocity). As a result, with this model, Debye shows how the heat capacity is directly related to the rate of change of the energy expectation value $\langle E \rangle$ with respect to temperature. Now $\langle E \rangle$ itself in turn depends on the  density of states $g(\omega)$ available per vibrational mode of a metal. ($n_B$ below is the Bose factor)
$$
\langle E \rangle = \int_0^\infty d\omega g(\omega)\hbar \omega(n_B(\beta \hbar \omega)+1/2) \tag{*}
$$
Now we are getting closer to where the Debye temperature or frequency (interchangeably used as they are related by a constant) comes into play. 
$$
g(\omega)\propto N\frac{\omega^2}{\omega_d^3}
$$
Remember that $g(\omega)$ simply tells you how many modes there are per frequency, this together with the energy of each mode, integrated over all allowed modes, gives the expected value of $E.$ Cutting the derivation short, the heat capacity can be shown to be:
$$
C=\frac{\partial \langle E \rangle}{\partial T} \propto N k_B \frac{T^3}{T_{Debye}^3}
$$
with $T_{Debye}=\frac{\hbar \omega_{Debye}}{k_B}.$
The problem with this model, as it stands, is that allows the heat capacity to grow indefinitely with $T^3,$ however from experimental results we know that the heat capacity drops off to $3k_B N$ at very high temperatures. ($N$ being the number of particles, remember that there should be as many modes as there are degrees of freedom, and not more)
Debye then corrected his model for this, by defining an ad hoc cut-off frequency (or temperature), which defines the highest allowed vibrational mode for a metal, which in turn bounds the behaviour of $C$ at very high $T.$ He then showed that this cut-off temperature is actually given by the Debye temperature.  With this now, the integral in $(*)$ is $\langle E \rangle=\int_0^{\omega_{Debye}}\dots$
From a practical point of view, you see now that the Debye temperature is directly to the heat capacity of a metal. In general hard materials have high Debye temperatures (e.g. Diamond), whereas e.g. lead (rather soft) has a low Debye temperature. Finally, in acoustic measurements, the speed of sound in a metal is directly linked to its Debye temperature.
