Graphene-based devices have important applications attributed to their superior performance and flexible tunability in practice. In this paper, a new kind of absorber with monolayer graphene sandwiched between two layers of dielectric rings is proposed. Two peaks with almost complete absorption are realized. The mechanism is that the double-layer dielectric rings added to both sides of the graphene layer are equivalent to resonators, whose double-side coupled-cavity effect can make the incident electromagnetic wave highly localized in the upper and lower surfaces of graphene layer simultaneously, leading to significant enhancement in the absorption of graphene. Furthermore, the influence of geometrical parameters on absorption performance is investigated in detail. Also, the device can be actively manipulated after fabrication through varying the chemical potential of graphene. As a result, the frequency shifts of the two absorption peaks can reach as large as 2.82 THz/eV and 3.83 THz/eV, respectively. Such a device could be used as tunable absorbers and other functional devices, such as multichannel filters, chemical/biochemical modulators and sensors.