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Bose-Einstein statistics govern systems of indistinguishable bosons with no restriction on state occupancy. The Bose-Einstein distribution g(E) = 1/(e(E-μ)/k_BT - 1) shows a singularity below the condensation temperature T_c, below which macroscopic numbers of particles occupy the ground state. This behavior explains Bose-Einstein condensation, superfluidity, and laser operation.
From your study of bosons and fermions, you know that identical quantum particles come in two flavors based on their spin: fermions (half-integer spin) obey the Pauli exclusion principle and can never share a quantum state, while bosons (integer spin) have no such restriction — any number of them can occupy the same state simultaneously. Bose-Einstein statistics is what happens when you take that permission seriously and count all allowed configurations of a gas of indistinguishable bosons.
The result is the Bose-Einstein distribution: the average number of bosons occupying a single-particle state with energy E is n(E) = 1 / [exp((E − μ)/k_BT) − 1], where μ is the chemical potential and T is temperature. Compare this to the Fermi-Dirac distribution for fermions, which has a +1 in the denominator instead of −1. That sign difference is everything. For fermions, n(E) is bounded above by 1 (exclusion principle). For bosons, n(E) is unbounded — the −1 in the denominator means that as E approaches μ from above, n(E) diverges. Bosons actively tend to pile into low-energy states, especially at low temperatures.
This tendency has a spectacular consequence at sufficiently low temperatures: Bose-Einstein condensation (BEC). Below a critical temperature T_c, the chemical potential reaches the ground-state energy, and a macroscopic fraction of all the bosons collapse into that single lowest-energy mode. This is not a classical phenomenon — it is driven entirely by quantum statistics. The condensed fraction behaves as a single coherent quantum state, which is why BECs exhibit superfluid behavior (flowing without viscosity) and laser-like coherence. Helium-4 becomes superfluid below 2.17 K for this reason, and dilute atomic BECs (achieved in 1995) allow direct experimental observation of the condensate.
Two other physical systems are described by the same Bose-Einstein counting. Photons in a cavity are bosons with μ = 0 (since photon number is not conserved), yielding the Planck distribution and blackbody radiation. Phonons — quantized lattice vibrations — are also bosons with μ = 0, and their Bose-Einstein distribution governs the heat capacity of solids (leading to the Einstein and Debye models). In each case, the key physics is the tendency of bosons to condense into low-energy modes, a tendency that becomes dramatically visible near absolute zero but shapes the thermodynamics of these systems at all temperatures.