Recently, metamaterials have attracted considerable attention because of their unique properties and potential applications in many areas, such as in bio-sensing, imaging, and communication. Among these researches, the metamaterial absorber has aroused much interest of researchers. The metamaterial absorber is important due to a broad range of potential application to solar energy, sensing, coatings for reducing the reflection, and selective thermal emitters. As a two-dimensional honeycomb structure composed of a single layer carbon atom, graphene is a promising candidate for tuning metamaterials and plasmonic structures due to its unique properties which differ substantially from those of metal and semiconductors. In this paper, we propose a tunable terahertz absorber based on graphene complementary metamaterial structure by removing periodic cut-wires on the graphene meta-surface. On the basis of the tunability of graphene conductivity, the absorber possesses a frequency tunable characteristic resulting from the change of graphene Femi level by altering the applied voltage. Here, we mainly study the influences of Fermi level of graphene and the size of the structure on the absorption characteristic of this metamaterial absorber. We finally obtain the corresponding Femi level and structural size under the perfect absorption condition. In addition, we utilize the multiple reflection theory to explore the physical mechanism, and verify the feasibility of the simulation method at the same time. The research indicates that the absorber can achieve 99.9% perfect absorption at 1.865 THz when the graphene Femi level is 0.6 eV, the thickness of substrate is 13 m, and the length and width of slit are 2.9 m and 0.1 m, respectively. When graphene Femi level increases from 0.4 eV to 0.9 eV, the resonance frequency of the absorber is blue-shifted from 1.596 THz to 2.168 THz. Meanwhile, the absorption rate increases from 84.68% at 0.4 eV to a maximum value of 99.9% at 0.6 eV, then gradually decreases to 86.63% at 0.9 eV. The maximum modulation of the absorption rate is 84.55% by varying the Femi level. When the thickness of substrate increases, the resonant frequency is red-shifted. The resonant frequency is blue-shifted when both the width and the length of the cut-wire on graphene increase. On the basis of the proposed graphene meta-surface absorber, one can gain different resonant frequencies by adjusting the structure geometric size and graphene Femi level. The graphene complementary structure can also be designed into different patterns to achieve the purpose of practical application.