What do we physically mean by smearing in condensed matter? What do we physically mean by term smearing in DFT based codes like VASP or generally in condensed matter? 
 A: The smearing in density functional theory codes means that you occupy the states of the Kohn-Sham system according to a smooth function, e.g., the Fermi distribution. It is introduced to avoid numerical problems, partly due to the finite sampling of the Brillouin zone and partly due to properties of the investigated system.
One example: Consider a metal with rather flat bands at the Fermi energy. If you have no temperature smearing, then such a band may be completely occupied if it is slightly below the Fermi energy, or completely unoccupied if it is slightly above the Fermi energy. Now consider your self-consistency cycle. In this cycle, the band can shift in energy from iteration to iteration. It can be occupied in one iteration and unoccupied in the next. This means that the charge density can change strongly from one iteration to the next, and such an effect may impede the convergence to a self-consistent solution. If you introduce a temperature smearing for the electronic system, then you obtain smoother changes of the occupations. The mixing algorithms that produce the next charge density from the charge densities of the previous iterations can deal better with such a situation.
That being said density functional theory is a ground-state theory. That means it is made for no temperature smearing of the electronic system. The self-consistent ground-state charge density that you want to obtain is the charge density for 0K. Thus the temperature smearing has to be seen as a convergence parameter. You may set it to a high temperature of 1000K or so to overcome larger convergence problems and then you go on and reduce it until your results don't change any more.
A: Many approximations or models result in discrete spectra, which are difficult to work with mathematically and numerically. E.g., periodic boundary conditions or any other boundary conditions that make the system finite (inevitable in numerics). Smearing then means converting these to a continuous spectrum, e.g., by assuming that the levels are broadened due to the processes not accounted in the model (e.g., electron-pnonon interaction, electron-electron scattering, etc.) The broadening is often taken to be of the order of temperature, which is more than just hand-waving - statistical phsyics provides us with solid arguments that we can neglect the residual interactions, while keeping in mind that they are responsible for the establishment of the thermodynamic equilibrium (characterized by temperature).
