A detailed analysis of electromechanically induced absorption (EMIA) in a circuit nano-electromechanical hybrid system consisting of a superconducting microwave resonator coupled to a nanomechanical beam is presented. By performing two-tone spectroscopy experiments, we have studied EMIA as a function of the drive power over a wide range of drive and probe tone detunings. We found good quantitative agreement between experiment and theoretical modeling based on the Hamiltonian formulation of a generic electromechanical system. We show that the absorption of microwave signals in an extremely narrow frequency band (Delta omega/2 pi < 5 Hz) around the cavity resonance of about 6 GHz can be adjusted over a range of more than 25 dB on varying the drive tone power by a factor of 2. Possible applications of this phenomenon include notch filters to cut out extremely narrow frequency bands (< Hz) of a much broader band of the order of MHz defined by the resonance width of the microwave cavity. The amount of absorption as well as the filtered frequency is tunable over the full width of the microwave resonance by adjusting the power and frequency of the drive field. At high drive power, we observe parametric microwave amplification with the nanomechanical resonator. Due to the very low loss rate of the nanomechanical beam the drive power range for parametric amplification is narrow, since the beam rapidly starts to perform self-oscillations.