The electron spin relaxation times measured in ESR spectroscopy are physically distinct from the electron spin relaxation times which appear in the theory of NMR Paramagnetic Relaxation Enhancement (NMR-PRE). ESR involves decay of a perturbed spin density matrix toward thermal equilibrium, while in NMR-PRE measurements, the electron spin density matrix remains at thermal equilibrium throughout the NMR experiment. The pertinent spin relaxation involves the thermal decay of the time correlation functions, G(r)(tau)=[S-r(0).S-r(tau)] (r = x,y,z), of the spin components, quantities which describe the persistence in microscopic correlation of the spin motion in the thermal equilibrium sample. The decay of the G(r)(tau) is shown to be level-specific; i.e., G(r)(tau) is composed of a sum of contribution associated with individual eigenstates, each of which decays exponentially via a process that is uncoupled to the decay in other eigenstates. This behavior differs markedly from the decay of the nonequilibrium parts of a perturbed density matrix, which involves coupled degree of freedom of the electron spin system. An expression for the level-specific relaxation times has been derived in terms of Redfield matrix elements. This expression is valid for any S greater than or equal to 1 when the static spin Hamiltonian consists of Zeeman and zfs contributions of arbitrary magnitude. Simple closed-form expressions are given for level-specific relaxation times in the cylindrical and orthorhombic zfs limits for S = 1 and S = 3/2. The theory is used to interpret electron and nuclear spin relaxation for S = 3/2 with specific reference to high-spin Co(II), for which the zfs splittings are typically large. For this spin system, the presence of orthorhombic terms in the zfs tensor causes profound shortening of the electron spin relaxation times relative to the reference cylindrical zfs case and, in consequence, a comparably large rhombicity-induced depression of the NMR relaxation efficiency.