Using first-principles pseudopotential plane wave method, the energy, atomic geometry and electronic density of states of FCC Cu crystal and its (111), (110) and (100) surface models were calculated and analyzed. According to the calculated results of the surface energy, the structural stability of the Cu surfaces increases for Cu (110), Cu (100), Cu (111) surfaces successively. The relaxation extent of the surface atoms decreases successively with the increasing the number of the layers. For the inwards relaxation of the surface layer atoms, Cu (110) surface moves maximum, Cu (100) takes second place, Cu (111) surface moves least. It was found that the relaxation of the surface atom layers not only causes the change of geometrical structures of the surface models but also leads to the change of peak contour of density of states (DOS) of surface layer atoms comparing with crystal inside. The increment of the total energy caused by these change is the main reason of the surface energy. And that the Cu (110) surface having higher activity than that of Cu(111) and Cu(100) surfaces may be attributed to its apparent rising of the surface layer atoms DOS in the high energy level.