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Universality means systems with different microscopic details exhibit identical critical exponents when belonging to the same universality class, determined by symmetry and dimensionality. Near the critical point, correlations diverge as ξ ~ |T - T_c|^{-ν}, and observables scale as powers of |T - T_c|, independent of system-specific parameters.
From your study of percolation and critical phenomena, you know that near a continuous phase transition, physical quantities diverge or vanish as power laws in the reduced temperature t = (T − T_c)/T_c. The magnetization goes as |t|^β, the susceptibility as |t|^{−γ}, the correlation length as |t|^{−ν}, and so on. What is not obvious, and in fact astonishing, is that these exponents β, γ, ν, and their companions are *the same* for wildly different physical systems. The liquid-gas critical point has the same exponents as the ferromagnetic transition in uniaxial magnets, which has the same exponents as a class of binary mixtures and polymer solutions. They are all in the same universality class: the 3D Ising universality class, with β ≈ 0.326 and ν ≈ 0.630.
Universality is counterintuitive because we usually expect that microscopic details matter. A ferromagnet is made of quantum spins on a lattice; a fluid is made of molecules with complicated interaction potentials. Why should they agree to three decimal places on their critical exponents? The answer, which the renormalization group (RG) makes precise, is that near the critical point the correlation length ξ diverges, meaning fluctuations are correlated over arbitrarily long distances. When ξ is much larger than any microscopic scale (atomic spacing, spin spacing), the microscopic details become irrelevant — they are "washed out" by the collective fluctuations. What remains are only the large-scale features: the symmetry of the order parameter and the spatial dimension d. Two systems with the same symmetry and dimension flow to the same fixed point under RG and therefore have the same critical exponents.
The universality class is thus characterized by two integers (roughly): the symmetry group of the order parameter and the number of spatial dimensions. The Ising class (Z₂ symmetry, scalar order parameter) covers uniaxial magnets, fluids, and alloys. The XY class (U(1) symmetry, two-component order parameter) covers superfluids and certain magnets. The Heisenberg class (O(3) symmetry, three-component order parameter) covers isotropic ferromagnets. Lower dimensions tend to have different exponents from higher dimensions, and at the upper critical dimension d_c (d_c = 4 for Ising) the exponents take their mean-field values — the simple predictions of theories that ignore fluctuations.
From your percolation work, you saw that geometric connectivity transitions share critical exponents with thermal transitions. Percolation belongs to its own universality class (different from Ising), but the *framework* is the same: diverging correlation length, power-law scaling, and exponents that depend only on dimension. The full apparatus of scaling relations — hyperscaling, Widom scaling, Fisher scaling — links the exponents together so that only two are independent; the others follow algebraically. These relations hold within each universality class and provide consistency checks for both theory and experiment. Universality classes are one of the most beautiful organizing principles in theoretical physics: they reveal that the universe has far fewer "types" of critical behavior than the number of distinct materials might suggest.