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Goldstone's theorem states that when a continuous global symmetry is spontaneously broken, one massless scalar boson (Goldstone boson) appears for each broken symmetry generator. These bosons correspond to excitations along the flat directions of the potential. The theorem is exact in relativistic field theory but is evaded when the broken symmetry is gauged (Higgs mechanism).
Goldstone's theorem, proved by Goldstone, Salam, and Weinberg in 1962, is one of the cornerstone results of quantum field theory. Its statement is precise: if a quantum field theory has a continuous global symmetry group G that is spontaneously broken to a subgroup H (meaning the vacuum is invariant under H but not under all of G), then the theory contains dim(G) - dim(H) massless scalar particles, one for each broken generator. These are the Goldstone bosons (or Nambu-Goldstone bosons, honoring Nambu's earlier work).
The proof is elegant. Consider the conserved Noether current jmu associated with a broken symmetry generator. The charge Q = integral j0 d3x does not annihilate the vacuum: Q|0> != 0 (this is what "broken" means). This implies that Q|0> is a state with the same energy as the vacuum (because [H, Q] = 0 -- the Hamiltonian respects the symmetry) but different quantum numbers. Taking the Fourier transform shows that this state has zero momentum and zero mass -- it is a massless particle created by the current jmu from the vacuum. The matrix element <0|j^mu(0)|pi(p)> = i f_pi pmu is nonzero, where f_pi is the "decay constant" of the Goldstone boson.
The most important physical example is in QCD. The approximate chiral symmetry SU(2)_L x SU(2)_R (exact in the limit of massless up and down quarks) is spontaneously broken to SU(2)_V by the formation of a quark condensate <q-bar q> != 0. The three broken generators produce three Goldstone bosons: the pions (pi+, pi-, pi0). Because the quark masses are small but not zero, chiral symmetry is also explicitly broken, giving the pions small masses proportional to sqrt(m_q). The pion mass hierarchy (m_pi = 140 MeV << m_proton = 938 MeV) is a direct consequence of the Goldstone mechanism applied to the approximate chiral symmetry of QCD.
In the context of gauge theories, Goldstone's theorem is modified by the Higgs mechanism. When the broken symmetry is a local (gauge) symmetry rather than a global one, the Goldstone bosons do not appear as physical particles. Instead, they provide the longitudinal degree of freedom that a massless gauge boson needs to become massive. The counting works out: a massless gauge boson has 2 polarizations, a massive one has 3, and the extra polarization comes from the eaten Goldstone boson. This is how the W and Z bosons of the Standard Model acquire their masses.