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Coupling constants in quantum field theory are not fixed numbers but depend on the energy scale at which they are measured. The beta function governs this energy dependence. In QED, the coupling increases at higher energies (charge is anti-screened at short distances); in QCD, it decreases (asymptotic freedom). This scale dependence has profound physical consequences.
In classical physics, the electric charge of an electron is a fixed number. In quantum field theory, the effective charge depends on the distance (or equivalently, the energy) at which you measure it. This running of coupling constants is one of the most important consequences of quantum corrections. The physical mechanism in QED is vacuum polarization: virtual electron-positron pairs in the vacuum act as electric dipoles that screen the bare charge. At long distances (low energies), the screening is maximal, giving alpha approximately 1/137. At shorter distances (higher energies), you probe inside the polarization cloud and see a larger effective charge.
The running is governed by the beta function, defined as beta(g) = mu dg/dmu, where mu is the energy scale. A positive beta function means the coupling increases with energy; a negative one means it decreases. For QED, beta = 2 alpha2/(3pi) > 0 (at leading order), so the coupling grows logarithmically with energy: alpha(mu) approximately alpha(mu_0) / [1 - (2alpha(mu_0))/(3pi) ln(mu/mu_0)]. This predicts that alpha reaches the value 1/128 at the Z boson mass, in excellent agreement with experiment.
The physical consequences of running couplings are dramatic. In QCD (quantum chromodynamics), the beta function is negative due to gluon self-interactions, giving asymptotic freedom: the strong coupling alpha_s becomes small at high energies, making perturbative calculations reliable for hard scattering processes. At low energies, alpha_s grows large, and perturbation theory breaks down -- this is the regime of confinement, where quarks and gluons are permanently bound into hadrons. The transition from perturbative to non-perturbative QCD occurs at Lambda_QCD approximately 200 MeV, which sets the scale of hadronic physics.
The running of all three Standard Model gauge couplings can be extrapolated to high energies using the renormalization group equations. The remarkable (and experimentally verified) fact is that the three couplings, which are very different at low energies, approach each other at around 1015-1016 GeV. This near-convergence is suggestive of grand unification -- the hypothesis that all three forces merge into a single force at very high energies. Whether the couplings exactly converge (and if so, at what scale) depends on the particle content of the theory between the electroweak scale and the unification scale, making this one of the key tests for theories beyond the Standard Model.