Spontaneous symmetry breaking occurs when the Lagrangian has a symmetry but the vacuum state does not.
The Mexican hat (wine-bottle) potential: V(φ) = μ²|φ|² + λ|φ|⁴. When μ² < 0, the symmetric point φ=0 becomes unstable — the field rolls to the degenerate minimum |φ| = v = √(−μ²/2λ).
Goldstone's theorem: each broken continuous symmetry produces a massless boson. The Goldstone mode is motion along the valley (zero restoring force); the massive Higgs mode is radial oscillation into the valley wall.