This paper presents a microwave absorber with dual absorption bandwidth response based upon the direction of electromagnetic wave incident on the surface. The design unit cell comprises a staircase shape metallic patch on the top plane and an array of 2×2 meander square ring shape dual layer frequency selective surfaces (FSS) in the middle and bottom planes. The relative absorption bandwidth (RAB) of 39.40% (5 GHz) with more than 90% absorption of incident wave power is achieved when an electromagnetic wave impinges normally on the top plane making it suitable for wideband applications in the X and Ku bands. For the wave incident normally on the bottom plane, the same structure gives narrow band absorption with an RAB of 2.29% (260 MHz) for more than 90% absorption around 11 GHz. Thus, this bi-directional ability of the proposed design is found to be suitable for radar absorbing material, multi-bandwidth, and diverse applications. The absorption performance is also studied for different values of incident angle. The distribution of surface currents on the staircase patch and on the two FSS layers at resonant frequencies of 11 GHz and 14 GHz is analyzed to elaborate the absorption phenomenon physically. The prototype of this design is fabricated and the experimental results are found to be closely following the simulated one.