As many other dimensionless numbers in transport phenomena, the Nu number compares the strength of two mechanisms in the context of heat transfer, namely: Convection / Conduction. In practice, just as the Re number indicates how turbulent your flow is (its nothing but the ratio of advection over diffusion-viscosity effects), the Nu number indicates which heat transfer mechanism is dominant in your system.
Look at the terms in the definition of the Nu number:

Nu = h * L / k
h= convective heat transfer coefficient (W/m┬▓ K)
L = characteristic length in your system (m)
k: thermal conductivity (W/m K)

Depending on the application, there are different correlations (typically based on Re and Pr) that help you to estimate the Nu number. These correlations are derived for specific cases and take into account the boundary layer, momentum, viscosity effects, etc..
So at the end of the day you will have something like

Nu = f(Re,Pr) = h * L / k

from which you can derive the convective heat transfer coefficient (HTC). Please look at or read Incropera's book.