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The Debye model treats solid vibrations as a gas of phonons with a linear dispersion relation ω = v_s k up to a cutoff frequency ω_D. The density of states g(ω) = 9N ω^2 / ω_D3 for ω ≤ ω_D recovers the Einstein model limit at high T and gives C_V → 12π^4 R/5 (T/Θ_D)3 at low T.
From your study of heat capacity of gases, you know that the equipartition theorem predicts C_V = (f/2)R per mole for each quadratic degree of freedom. For a monatomic solid, each atom has three kinetic and three potential energy degrees of freedom, giving C_V = 3R — the Dulong-Petit law, which works well at high temperatures. But experiments show that heat capacity falls dramatically below 3R at low temperatures, eventually approaching zero as T → 0. The Debye model is the quantum statistical mechanics story that explains this falloff.
The key physical picture is that atoms in a solid don't vibrate independently — they are coupled, and their collective vibrations form waves that travel through the crystal. These quantized sound waves are called phonons, and they play the same role for lattice vibrations that photons play for electromagnetic radiation. At low temperatures, most high-frequency vibrational modes are "frozen out" because thermal energy k_BT is too small to excite a phonon of energy ℏω. Only low-frequency, long-wavelength phonons get excited, and there are few of them — hence the low heat capacity. The partition function approach you may have encountered makes this precise: each phonon mode contributes to heat capacity only when k_BT ≳ ℏω_mode.
The Debye model's improvement over the Einstein model (which treated all atoms as independent oscillators at a single frequency) is in the density of states. Real phonons have a range of frequencies from zero up to a maximum Debye frequency ω_D, with a density of states g(ω) ∝ ω². This quadratic density of states reflects the geometry of three-dimensional wave propagation — just as in electromagnetic radiation, lower frequencies crowd together more densely in frequency space. The cutoff ω_D is set by requiring that the total number of modes equal 3N (three vibrational modes per atom), fixing ω_D in terms of the speed of sound and the atomic density.
The two limiting regimes are clean and physically transparent. At high temperature (k_BT >> ℏω_D), all modes are thermally excited and equipartition holds: C_V → 3R, recovering Dulong-Petit. At low temperature (k_BT << ℏω_D), only the low-frequency ω ∝ k modes near the origin are populated, and the calculation gives the celebrated Debye T³ law: C_V ∝ (T/Θ_D)³, where the Debye temperature Θ_D = ℏω_D/k_B characterizes the material. Diamond, with its stiff bonds and light carbon atoms, has Θ_D ≈ 2200 K — its modes are hard to excite, and its room-temperature heat capacity is well below 3R. Lead, with heavy atoms and weak bonds, has Θ_D ≈ 100 K — nearly all its modes are active at room temperature.