The structural and magnetic properties of ${\mathrm{Co}}_{18\ensuremath{-}m}{\mathrm{Cu}}_{m}$ $(0l~ml~18)$ clusters are investigated with a genetic algorithm and a $\mathrm{spd}$-band model Hamiltonian in the unrestricted Hartree-Fock approximation, respectively. In general, Cu atoms tend to occupy the surface, while Co atoms prefer to the interior of the clusters. Layered structures appear in some clusters with given stoichiometric compositions. The introduction of Cu atoms leads to a large increase of the magnetic moment of Co-rich circumstance and nearly zero magnetism of the Cu-rich ambient. The interaction between Cu and Co atoms induces nonzero magnetic moment for Cu atoms. The total magnetic moments tend to decrease with the increase of Cu atoms. However, some particular large magnetic moment are found to be closely related to the structures. The environments of Cu and Co atoms have a dominant effect on the magnetism of the cluster.