Nonlocal dielectric function - what does it mean? I'm reading this* article and in the second sentence of introduction I encountered a term I haven't heard of before. Namely: nonlocal dielectric function. What does this nonlocality mean? And how does it differ from locality? What is its physical meaning?
*R.Ruppin Phys Rev B vol 45 nr 19 page 11 209.
 A: It means polarization $\mathbf P$ at point $\mathbf x$ is an integral involving electric field $\mathbf E$ at points $\mathbf x'$ different from $\mathbf x$.
A: To add a little more detail to Ján's correct answer, the polarization at a point is not simply proportional to the electric field at that point but rather depends on the electric field over a nonzero volume. For linear materials, this effect can be modelled by a spatial "impulse response" or kernel $\chi(\vec{x})$, which is a matrix-valued function of position:
$$\vec{P}(\vec{x}) = \int_{\mathbb{R}^3} \chi(\vec{x} - \vec{x}^\prime)\,\vec{E}(\vec{x}^\prime)\,\mathrm{d}^3 x$$
and $\chi$ is a simple scalar if the medium is isotropic. In all practical situations, the kernel $\chi$ has spatial frequency components that are as high as or less than the wavenumber, reflecting the fact that the field of influence of a propagating electromagnetic field spreads over regions of the order of a wavelength or wider in breadth. 
An experimental effect of this equation is that, even in the "empty" spaces between atoms in a solid lattice, the electromagnetic field behaves as though there is polarizable material there. In more everyday language: "photons can't see structure finer than a wavelength very well; fine structure averages out over wavelength-sized regions".
A: I found a paper that discusses this nonlocality in the context of intermolecular interactions.  Not the same context as the article you linked, but it may help with understanding the concept.
"The nonlocal dielectric function of a molecule determines the effective potential at a certain point due to an applied external potential at a different point, within the molecule. The effective potential at point r is determined by the nonlocal dielectric function ϵv ( r, r'; ω) within linear response and by the nonlocal dielectric function ϵq, (r, r', r"; ω, ω') within nonlinear response to the lowest order." - http://gradworks.umi.com/33/96/3396071.html
The abstract continues: "...the nonlocal dielectric model gives the interaction energies and forces at second and higher-orders, in agreement with the results from quantum mechanical perturbation theory."  The paper is behind a paywall, but you may be able to access it free if your library subscribes.  If not, you may be able to get a 24-page preview free.
A: Just to add, nonlocal dielectric response also leads to the permittivity being dependent on the wavevector. Nonlocality is thus tightly bound to the notion of spatial dispersion. 
This has profound implications on the light propagation is nonlocal media. The dispersion curves can be bent upwards or downwards with the wavenumber. Therefore, it may occur that two parallel, but independent waves with different wavenumber can propagate at the same frequency. Such spatial dispersion may also result in opposite signs of the group and phase velocities, leading to negative refraction without index of refraction being even defined. (Nice discussion is e.g. given by Agranovich in https://mesoscopic.mines.edu/mediawiki/images/a/ab/Spatial_dispersion.pdf)
Spatial dispersion is quite common in natural media, but usually it is weak. Some effects that seem as classical frequency dispersion are, in fact, a manifestation of a weak spatial dispersion, for example the Doppler line broadening in gases [c.f. Landau & Lifshitz, Continuum Electrodynamics]. Spatial dispersion however can become very pronounced when metamaterials or photonic crystals are to be treated as homogeneous media. 
