Pseudo-Goldstone bosons arise in a quantum field theory with both spontaneous and explicit symmetry breaking. The controlling approximate symmetries, if they were exact, would be spontaneously broken (hidden), and would thus engender massless Goldstone bosons. The additional explicit symmetry breaking gives these bosons a small mass. The properties of these pseudo-Goldstone bosons can normally be found by an expansion around the (exactly) symmetric theory in terms of the explicit symmetry-breaking parameters.
Quantum chromodynamics (QCD), the theory of strong particle interactions, provides the best known example in nature; see the article on the QCD vacuum for details. Experimentally, it is observed that the masses of the octet of pseudoscalar mesons (such as the pion) are much lighter than the next heavier states, i.e., the octet of vector mesons (such as the rho meson).
In QCD, this is interpreted as a consequence of spontaneous symmetry breaking of chiral symmetry in a sector of QCD with 3 flavours of light quarks. Such a theory, for idealized massless quarks, has global chiral flavour symmetry. Under SSB, this is spontaneously broken to the diagonal SU(3), generating eight Goldstone bosons, which are the pseudoscalar mesons transforming as an octet representation of flavour SU(3).
In actual full QCD, the small quark masses further break the chiral symmetry explicitly as well. The masses of the actual pseudoscalar meson octet are found by an expansion in the quark masses, which goes by the name of chiral perturbation theory. The internal consistency of this argument is further checked by lattice QCD computations, which allow one to vary the quark mass and check that the variation of the pseudoscalar masses with the quark masses is as dictated by chiral perturbation theory, effectively as the square-root of the quark masses.