We have investigated the electronic structures and magnetism of a full Heusler alloy $Co_2CrGa(001)$ surface by using the all-electron full-potential linearized augmented plane wave (FLAPW) method within the generalized gradient approximation (GGA). We considered two types of different terminations: the Co-terminated (Co-Term) and the CrGa-terminated (CrGa-Term) surfaces. From the calculated layer-projected density of states (LDOS), we found that the surface of the CrGa-Term shows nearly half-metallic character while that of the Co-Term is far from the half-metallic. For the Co-Term, the surface Co atom moves down to the bulk region by $0.05{AA}A$, while the subsurface Cr and Ga atoms move up to the surface layer by 0.05 and $0.01{AA}$, respectively. For the CrGa-Term, there is a large inward relaxation of the surface Ga atom $(0.07{AA})$, but the relaxation of the surface Cr atom is very small $(0.01{AA})$. The relaxations affect not much to the overall shapes of DOS for both terminations, but make the surface states of the surface Cr and Ga atoms for the CrGa-Term shift to higher energy that enhances the nearly half-metallic character of the CrGa-Term. The magnetic moments of the surface $Cr(2.98{mu}_B)$ in the CrGa-Term and the surface $Co(1.17{mu}_B)$ in the Co-Term were much increased compared to those of the inner-layers $(1.79;and;0.77{mu}_B)$, respectively, while that of the subsurface Cr atom in the Co-Term was decreased to $1.19{mu}_B$.