In this paper, a TM polarization multi-band absorber is achieved in a graphene-Ag asymmetrical grating structure. The proposed absorber can achieve perfect absorption at 1108 nm, 1254 nm, and 1712 nm (the absorption exceeds 98.4% at the three peaks). Results show that the perfect absorption effect originates from the excitation of magnetic polaritons (MPs) in the silver ridge grating; a LC equivalent circuit model is utilized to confirm the finite-difference-time-domain (FDTD) simulation. The influences of the incident angle, polarization angle, and geometrical size on the absorption spectrum are investigated. Moreover, a quadruple band absorber and a quintuple band absorber are also designed by introducing more silver grating ridges in one period. The proposed graphene-Ag asymmetrical structure has some advantages compared with other absorbers such as the ability to be independently tuned and a simple structure. Thus, the proposed structure can be applied in the areas of multiple absorption switches, near-infrared modulators, and sensors.