Definition
Gauge symmetry is invariance of a physical theory under transformations that can vary smoothly with position in spacetime. Requiring such local symmetry forces new fields — the gauge fields — into the theory, and these gauge fields turn out to be the carriers of fundamental forces.
The Standard Model's three non-gravitational forces all arise from gauge symmetries: U(1) for electromagnetism, SU(2) for the weak force, and SU(3) for the strong force.
Why it matters
How it works
Start with a quantum field — say, the electron field ψ. The free theory is invariant under a global phase rotation ψ → e^(iα) ψ for any constant α. Now demand the same invariance even when α varies with position, α(x). The kinetic terms of the theory no longer cancel, because derivatives now generate extra pieces.
The fix is to introduce a new vector field A and a covariant derivative that combines the ordinary derivative with A. Choose how A transforms so that the extra pieces cancel exactly. The price of local symmetry is the existence of A — and A turns out to be the photon. Electromagnetism is forced on you by requiring local U(1) symmetry.
The same procedure with a non-abelian group like SU(2) or SU(3) yields the W and Z bosons (after symmetry breaking) and the gluons. Non-abelian gauge fields interact with each other as well as with matter — a feature absent in electromagnetism that makes the strong force confine quarks and the weak force decay neutrons.
Gauge symmetries also constrain interactions: only certain terms in the Lagrangian preserve the symmetry, which dramatically limits what couplings can appear and predicts the existence of specific particles before they were observed. The W, the Z, and the gluon were all predicted from gauge structure and then found.