**Normal modes**  
Let us first take a classical view of coupled oscillators - e.g., balls connected by springs or atom in a lattice (which, for small displacements, can be reduced to the balls coupled by springs.) Classical mechanics teaches us that we can find the normal oscillation modes of such a system, and represent any oscillation as a superposition of such [*normal modes*][1]. Any single mode then obeys a harmonic oscillator equation:
$$
\ddot{Q}_k(t)+\omega_k^2Q_k(t)=0,\\
H_k=\frac{P_k^2}{2}+\frac{\omega_k^2Q_k^2}{2}.
$$
[See here][2] for an example of a dispersion relation that we can get:  
[![enter image description here][3]][3]

**Quantization**  
If we consider the same problem from the point of view of quantum mechanics, we expect that the excitations will be quantized. The quantization of Harmonic oscillator is a well-known problem - each mode will have discrete energies:
$$
E_k=\hbar\omega_k\left(n_k+\frac{1}{2}\right),
$$ 
where $n_k$ characterizes the level of excitation of modes. Note that, given the finite crystal size (i.e., the finite number of atoms - say $N$ atoms) we have finite number of modes, $k_1,...,k_N$, but the excitation numbers can be arbitrary.

**Second quantization**  
We now can perform second quantization or, in a simpler and oscillator-specific language, describe the modes in terms of [*raising and lowering operators*][4]:
$$
H_k=\hbar\omega_k\left(a_k^\dagger a_k+\frac{1}{2}.
$$
Note that the number of the oscillators is still $N$, but the operators now work in the Fock space (filling numbers space), so the level of excitations can be arbitrary.

**Phonons**  
The last step is calling these new excitations - *phonons*. In this case, $a_k^\dagger, a_k$ are no more raising and lowering operators, but phonon creation and annihilation operators, whereas $n_k$ becomes a *number of phonons* excited in mode $k$. In many ways phonons behave just any other particles, although for the sake of caution one calls them *quasiparticles*. There exist many other similar excitations, which are obtained from oscillatory modes - plasmons (for electron plasma in metals), magnons (for spin waves), polaritons (in Bragg lattices or cavities), etc.

**Related**  
[*Is differentiating particle and quasiparticle meaningless?*][5]  
[*Do quasiphotons have mass?*][6]

**Reading**  
Charles Kittel, [*Introduction to solid state physics*][7] - standard text  
H. Haken, [*Quantum field theory of solids*][8] - an accessible introduction to solid state physics from quantum quasiparticles viewpoint


  [1]: https://en.wikipedia.org/wiki/Normal_mode
  [2]: https://en.wikipedia.org/wiki/Phonon#Three-dimensional_lattice
  [3]: https://i.sstatic.net/bm73DuNUm.png
  [4]: https://en.wikipedia.org/wiki/Ladder_operator
  [5]: https://physics.stackexchange.com/q/604029/247642
  [6]: https://physics.stackexchange.com/q/751290/247642
  [7]: https://en.wikipedia.org/wiki/Introduction_to_Solid_State_Physics
  [8]: https://books.google.fr/books/about/Quantum_Field_Theory_of_Solids.html?id=QGosAAAAYAAJ&redir_esc=y