In hindsight, I suppose that I should have addressed phonons earlier. A phonon is a quantized sound wave - a collective vibrational mode of a solid (or liquid). In a crystalline solid, the idea is that the atoms in the solid are displaced, at any given instant, from their equilibrium positions. For a single phonon, the instantaneous displacements are periodic in space (that is, there is some wavelength, where atoms separated by an integer number of wavelengths are displaced the same amount). The displaced atoms feel restoring forces due to their interactions with neighbors, and will tend to oscillate in time around their equilibrium positions. When the wavelength is much longer than the interparticle separation, the frequency of those oscillations times the wavelength gives the speed of sound for the material - phonons propagate along at the speed of sound. In general, the speed of sound can depend on the direction of propagation as well as the direction of the direction of the displacement. If the displacement is along the direction of propagation, the sound is longitudinal; if the displacement is normal to the direction of propagation, the sound is transverse.
The quantum nature of phonons comes in when one discusses their energy content. In a classical mechanical oscillator, you can dump in as much energy as you want; the energy content is proportional to the square of the amplitude of the oscillation, and that can be varied continuously. In a quantum mechanical oscillator of frequency f, the energy content of that oscillator can only take on discrete values, (n + 1/2)hf, where n is a nonnegative integer. This is a subtle yet hugely important distinction. Mathematically it explains a major contribution to the heat capacity of crystalline solids at low temperatures (and it's very strongly related to the form of blackbody radiation when one is worrying about photons rather than phonons).
Because they have a wavelength and therefore a wavevector (and an effective momentum) as well as an energy, one can think about processes that involve the emission, propagation, and scattering of phonons - they have particle-like attributes in that sense.
(For a layperson discussion, I'm avoiding subtle distinctions like acoustic vs. optical phonons. If you really care, in acoustic phonons all the atoms within a unit cell move together, while for optical phonons different atoms within a single unit cell move by different amounts.)