We give a discrete analogueof the harmonic morphism between two Riemannian manifolds. Roughly speaking, this is a mapping betweentwo graphs preserving local harmonic functions. We characterize harmonic morphisms in terms ofhorizontal conformality. Many examples including coverings, non-complete extended p-sums andcollapsings are given. Introducing the horizontal and vertical Laplacians, the Green kernel estimatesare obtained for the harmonic morphism. As applications, a general and sharp estimate of the Greenkernel for an infinite tree is obtained.