Droplets impinging on solid surfaces is a common phenomenon. However, the motion of surfaces remarkably influences the dynamical behaviors of droplets, and related research is scarce. Dynamical behaviors of water nanodroplets impinging on translation and vibrating solid copper surfaces were investigated via molecular dynamics (MD) simulation. The dynamical characteristics of water nanodroplets with various Weber numbers were studied at five translation velocities, four vibration amplitudes, and five vibration periods of the surface. The results show that when water nanodroplets impinge on translation surfaces, water molecules not only move along the surfaces but also rotate around the centroid of the water nanodroplet at the relative sliding stage. Water nanodroplets spread twice in the direction perpendicular to the relative sliding under a higher surface translation velocity. Additionally, a formula for water nanodroplets velocity in the translation direction was developed. Water nanodroplets with a larger Weber number experience a heavier friction force. For cases wherein water nanodroplets impinge on vibration surfaces, the increase in amplitudes impedes the spread of water nanodroplets, while the vibration periods promote it. Moreover, the short-period vibration makes water nanodroplets bounce off the surface.