Normal sound waves are fluctuations in the displacement and density of molecules in a substance;
second sound waves are fluctuations in the density of quasiparticle thermal excitations (rotons and phonons). Second sound can be observed in any system in which most phonon-phonon collisions conserve momentum, like superfluids and in some dielectric crystals when Umklapp scattering is small. 
Contrary to molecules in a gas, quasiparticles are not necessarily conserved. Also gas molecules in a box conserve momentum (except at the boundaries of box), while quasiparticles can sometimes not conserve momentum in the presence of impurities or Umklapp scattering. Umklapp phonon-phonon scattering exchanges momentum with the crystal lattice, so phonon momentum is not conserved, but Umklapp processes can be reduced at low temperatures. 
Normal sound in gases is a consequence of the collision rate  between molecules being large compared to the frequency of the sound wave . For second sound, the Umklapp rate  has to be small compared to the oscillation frequency  for energy and momentum conservation. However analogous to gasses, the relaxation time  describing the collisions has to be large with respect to the frequency , leaving a window:
for sound-like behaviour or second sound. The second sound thus behaves as oscillations of the local number of quasiparticles (or of the local energy carried by these particles). Contrary to the normal sound where energy is related to pressure and temperature, in a crystal the local energy density is purely a function of the temperature. In this sense, the second sound can also be considered as oscillations of the local temperature. Second sound is a wave-like phenomena which makes it very different from usual heat diffusion.