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.
There’s more, though.
Also see What is a plasmon?