The potential energy curves and transition dipole moments of 12Σ+, 22Σ+, 12Π and 22Π states of NH+ cation and NH− anion are calculated by using multi-reference configuration interaction method and large all-electron basis sets. Based on the obtained potential energy curves, the rotational and vibrational energy levels of the states are obtained by solving the Schrödinger equation of nuclear movement. The calculated spectroscopic parameters for NH+ cation and NH− anion are in good agreement with available theoretical and experimental results. The spin orbit coupling effect of the 2Π states for both NH+ cation and NH− anion are calculated. The feasibility of laser cooling of the two molecules is examined by using the results of the molecular structure and spectroscopy. The highly diagonal Franck-Condon factors for the 12Π (v″=0)↔12Σ+ (v′=0) transition of NH+ and NH− are 0.821 and 0.999, while the radiative lifetimes of the 12Σ+ (v′=0) state for the two molecules are 384ns and 52.4ns, respectively. The results indicate that NH+ cation and NH− anion are good candidate molecules for laser cooling. The cooling scheme via Sisyphus process for the NH+ cation and NH− anion are proposed in the paper. The laser wavelengths for the close cycles of the absorption and radiation are also determined. Unfortunately, the potential energy curve of the ground state of the neutral NH molecule shows that the auto-detachment of NH− anion is possible, implying the optical scheme of laser cooling for NH− anion is not easy to achieve in the experiment although it has larger Franck-Condon factor.