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Below the critical temperature, a macroscopic number of bosons occupy the ground state, and the system acquires a non-zero order parameter: the condensate wavefunction ψ. This spontaneous breaking of gauge symmetry is the hallmark of a quantum phase transition and manifests in phenomena like superfluidity.
From your study of the ideal Bose gas, you know that bosons do not obey the Pauli exclusion principle — any number of them can occupy the same single-particle quantum state. The Bose-Einstein distribution nₖ = 1/(exp((εₖ − μ)/kT) − 1) describes the average occupation. For the chemical potential to remain negative (so that occupation numbers stay positive), μ must satisfy μ < ε₀ = 0 (the ground state energy). As temperature decreases, μ increases toward zero. At the critical temperature Tc, μ hits zero — and the ground state occupation n₀ = 1/(exp(−μ/kT) − 1) becomes macroscopic. Below Tc, a finite fraction of all N particles pile into the ground state. This macroscopic ground state occupation is Bose-Einstein condensation.
The phrase "macroscopic occupation" is key. In a normal gas at temperature T, each single-particle state has of order 1 particle on average (or much less). In the condensate, the ground state has of order N particles — a number proportional to the system size. This is a qualitative distinction. You can estimate Tc from the condition that the thermal de Broglie wavelength λ_dB becomes comparable to the interparticle spacing: nλ_dB³ ≈ 2.612. When particles are close enough together that their wavefunctions overlap significantly, they "feel" the bosonic statistics and begin to collectively pile up.
The connection to your prerequisite on order parameters is subtle but important. In a standard phase transition, the order parameter is a classical object (average magnetization, average density difference). For BEC, the condensate wavefunction ψ = √(n₀) eiφ plays this role — it is a complex number, with both a magnitude (the square root of condensate density) and a phase φ. Above Tc, ψ = 0. Below Tc, ψ ≠ 0 and a definite phase is chosen, spontaneously breaking the U(1) gauge symmetry (the symmetry ψ → eiαψ). This is a quantum phase transition: the order parameter is literally a quantum mechanical wavefunction that has become macroscopic.
The physical consequences are dramatic. When a macroscopic number of particles share the same quantum state, they also share the same phase — they are phase coherent. This coherence is what enables superfluidity: the condensate flows without viscosity because there is no mechanism to scatter it into a different phase-coherent state without paying a large energy cost. The same physics appears in superconductivity (where Cooper pairs of electrons condense) and in laser light (photons in a coherent state). Bose-Einstein condensation was first achieved experimentally in ultracold atomic gases in 1995, confirming predictions made by Einstein in 1924, and has since become a leading platform for studying quantum many-body physics in controlled laboratory settings.