In order to reduce the size of the device and realize the ultrafast response time and dynamic tunableness, the single-band and dual-band plasmon induced transparency (PIT) effect are investigated based on graphene nanoribbon waveguide side-coupled rectangle cavity. The slow light properties of the model are analyzed numerically and theoretically by coupled mode theory and finite difference time domain method. With controlling the chemical potential of the graphene rectangle cavity, the tunability of the resonant wavelength and the transmission peak can be achieved simultaneously in single-band and dual-band PIT model. As the chemical potential of graphene increases, the resonant wavelength of each transmission window of PIT effect decreases gradually and presents the blue shift. In addition, through dynamically tuning the resonant wavelength of the graphene rectangle cavity, when the chemical potential of the graphene rectangle cavity increases from 0.41 to 0.44 eV, the group index of single PIT system is controlled to be between 79.2 and 28.3, and the tunable bandwidth is 477 nm. Moreover, the group index of dual PIT system is controlled to be between 143.2 and 108.6 when the chemical potentials of graphene rectangle cavities 1, 2, and 3 are 0.39–0.42 eV, 0.40–0.43 eV, and 0.41–0.44 eV, respectively. The size of the entire PIT structure is <0.5 μm². The research results here in this work are of reference significance in designing and fabricating the optical sensors, optical filters, slow light and light storage devices with ultrafast, ultracompact and dynamic tunableness.