This generalizes the classical situation when the manifold is a flat Euclidean space. Harmonic maps are solutions to an elliptic system of partial differential equations, which in general is non-linear.
Harmonic maps are very important both in classical and modern differential geometry.
The best known application is their parametrization of both geodesics and minimal surfaces in Riemannian manifolds. J. Eells, L. Lemaire, A report on harmonic maps , Bull. J. Eells, L. Lemaire, Selected topics in harmonic maps , CBMS Regional Conf. Ser. in Math. J. Eells, L. Lemaire, Another report on harmonic maps , Bull.