The relative motion between a magnet and a coil generates a current in the coil. This phenomenon is called induction. It is irrelevant whether the magnet or the coil is being moved. In Maxwell's theory of electromagnetism there are, however, two entirely different explanations for the causes, in dependence of whether the coil or the magnet is being moved.
Einstein points out this asymmetry in the beginning of his 1905 paper on the Electrodynamics of Moving Bodies in which he develops the Theory of Special Relativity. In a later account of 1920 he also emphasizes the decisive role of this asymmetry in which a borderline problem of mechanics and electrodynamics manifests itself.