As well as Born&Wolf, you might like to look at Chapter 11 of volume 2 of the "Feynman lectures on physics", called "inside dielectrics". This is an excellent beginner's classical description of the mechanisms behind a material's polarisability.
Actually it's not the $damped$ motion of electrons that gives rise to the usual, real valued permittivity. Damping (i.e. energy loss) begets the imaginary part of permittivity and leads to the attenuation of light as it propagates through the material. The permittivity's real part simply arises from light's being "slowed down" by the material as follows. A photon is propagating through free space and is then absorbed by one of the material's constituent (atom or molecule), fleetingly raising the latter into a higher energy state. A very short time later, this excited constituent emits a photon in the same direction as the first. The second one propagates on, is absorbed by another matter constituent, and a third is emitted a short time later, again with exactly the same momentum as the second. This repeated absorption, delay, emission can be modelled by saying the original photon is just running slower than it would in free space. Call me pedantic, but I like to say that it's not light that is propagating through the material, but a quantum superposition of photons and excited matter states.
This process is somewhat like fluorescence, but with two crucial differences:
It is $extremely$ fast: the delay between absorption and re-emission is femtoseconds at the very most.
The absorber returns $exactly$ to its initial state and gains no energy nor momentum, so there is no wavelength shift and no direction change between the absorbed and re-emitted photons. Mostly there is no angular momentum transfer either, and this means that the light's polarisation stays the same - see the chapter "Angular Momentum" in volume 3 of the Feynman lectures to understand the relationship between polarisation and angular momentum. The exception to the last point is, of course, in a birefringent material.
Damping arises when the excited state is weakly coupled to a band of other, usually vibrational states in the matter. The weak coupling means that mostly photons are absorbed and re-emitted loss mostly as described above, but every so often the excited state couples to the vibration states and the photon concerned is thus permanently lost.
The lossless (real-valued) permittivity as a function of wavelength is described by the Sellmeier equations. With loss, these become the Ketteler–Helmholtz model. I have a couple of papers on this subject:
Vance RWC, Ladouceur F. One photon electrodynamics in optical fiber with fluorophore systems. II. One-polariton propagation in matter and fibers from the one-photon correspondence principle. J Opt Soc Am 2007; 24(4): 942-958.
Vance RWC, Ladouceur F. One photon electrodynamics in optical fiber with fluorophore systems. III. One-polariton propagation in fluorophore-driven fibers. J Opt Soc Am 2007; 24(6): 1369-1382.
A further paper aside from my own on the full Ketteler–Helmholtz model is:
G. Dubost and A. Bellossi, “Modèle physique de la membrane cellulaire dans les spectres infrarouge, visible etultraviolet,” Rev. Sci. Tech. Déf. 64, 113–127 (2004).
I'm really sorry: I haven't come across an english language description - but I notice Carlos has just given a good classical description in his answer. Also, as advised by Ben Cromwell, see physics.stackexchange.com/q/65812/4552