The crack in classical counting
In Statistical Mechanics I you counted microstates as if each particle wore a name tag — swap particle A and particle B and you claimed a different microstate. That innocent assumption breaks the theory: it makes entropy non-extensive and predicts a spurious change when you remove a partition between two identical gases. This is the Gibbs paradox.
The empirical patch is to divide the number of configurations by $N!$ — the number of ways to relabel identical particles. With that factor, entropy becomes extensive again and the paradox vanishes, as the Sackur-Tetrode equation shows.
The Sackur-Tetrode entropy of an ideal gas. The 1/N! from indistinguishability is exactly what makes S depend on the density V/N (an intensive quantity) rather than on V and N separately — restoring extensivity.
When does quantum-ness matter? The thermal wavelength
Every particle has a quantum fuzziness set by its de Broglie wavelength. At temperature T the typical momentum is thermal, and the corresponding wavelength is the thermal de Broglie wavelength \lambda_{\text{th}}.
The thermal de Broglie wavelength: colder or lighter particles are 'fuzzier'. It grows without bound as T → 0.
The decisive question is whether these quantum clouds overlap. Compare \lambda_{\text{th}} to the mean interparticle spacing n^{-1/3}, where n is the number density. The dimensionless degeneracy parameter is n\lambda_{\text{th}}^{3}.
The single number that decides everything. When the clouds barely overlap the gas is classical; when they crowd together, quantum statistics take over.
Two tribes: bosons and fermions
Because identical particles are truly interchangeable, swapping any two must leave all physics unchanged. The joint wavefunction can therefore only pick up a phase under exchange, and consistency forces that phase to be \pm 1.
The exchange operator gives +1 for bosons (symmetric wavefunction) and −1 for fermions (antisymmetric). The antisymmetry forbids two fermions from sharing a state — the Pauli principle in one line.
The spin-statistics theorem ties this to spin: integer-spin particles (photons, phonons, He-4, W and Z bosons) are bosons; half-integer-spin particles (electrons, protons, neutrons, He-3) are fermions. From the antisymmetry follows the Pauli exclusion principle: at most one fermion per single-particle quantum state.
The big picture in one chart
The whole track lives in one picture: how many particles occupy a state of energy \varepsilon, at temperature T. Fermions refuse to double up, so their occupancy never exceeds one; bosons love to crowd, so theirs can grow without limit; classical particles sit in between. At high energy or high temperature all three curves collapse onto the classical Maxwell-Boltzmann result.
Everything ahead is a consequence of these three curves: blackbody radiation and the specific heat of solids from the Bose curve; the stability of metals, white dwarfs and neutron stars from the Fermi step; and Bose-Einstein condensation from what happens when bosons run out of excited states to occupy.