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The electronic structures of Co/Cu interface have been calculated by first-principles method based on local spin density approximation. Models 3Co/xCu (x=1-8 monolayers) with different Cu layer thickness are investigated. The results show that the oscillation of the density of states near the Fermi surface with the Cu spacer thickness has been observed and the period of oscillation is about 6 atom layers, which has a good agreement with the corresponding experiments. We also discuss the spin polarization and magnetic resistance with the change of Cu layers thickness. Further analysis shows majority spin states near the Fermi surface played a key role in giant magnetoresistance (GMR).