In this paper, three Fano resonances based on three different physical mechanisms are theoretically and numerically investigated in a plasmonic resonator system, comprised of two circular cavities. And the multimode interference coupled mode theory (MICMT) including coupling phases is proposed to explain the Fano resonances in plasmonic resonator system. According to MICMT, one of the Fano resonances originates from the interference between different resonant modes of one resonator, the other is induced by the interference between the resonant modes of different resonators. Mode degeneracy is removed when the symmetry of the system is broken, thereby emerging the third kind of Fano resonance which is called degenerate interference Fano resonance, and the degenerate interference coupled mode theory (DICMT) is proposed to explain this kind of Fano resonance. The sensitivity and FOM* (figure of merit) of these Fano resonances can be as high as 840 nm/RIU and 100, respectively. These are useful for fundamental study and applications in sensors, splitters and slow-light devices.