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Equipartition usually refers to the fact that

in classical statistical mechanics each degree of freedom that appears quadratically in the energy (Hamiltonian) has an average value of \frac{1}{2}k_B T, where k_B T is the thermal energy.

Thus, the thermal energy is shared equally ("equipartitioned") by all these degrees of freedom. This is a consequence of the equipartition theorem, which is very simple mathematically. As an immediate corollary, the translational energy of a molecule must equal \frac{3}{2}k_B T, since translations are described by three degrees of freedom.

For elastic waves, equipartition refers to the fact that the average potential and kinetic energies are equal (and therefore equal to half the total energy, which is thereby "equipartitioned".)