Integrating gate-tunable graphene with plasmonic nanostructures or metamaterials offers a great potential in achieving dynamic control of plasmonic response. While remarkable progress has been made in realizing efficient graphene-induced modulations of plasmon resonances, a full picture of graphene-plasmon interactions and the consequent deep understanding on graphene-enabled tuning mechanism remain largely unexplored. Here, we theoretically identify, for the first time, two distinct modulation effects that can coexist in graphene-based plasmonic nanostructure: graphene can influence the plasmon resonances by either acting as equivalent nanocircuit elements or effectively altering their excitation environment, leading to totally different tuning behaviors. A general dependency of tuning features on the graphene-induced impedance, irrespective of structure geometries, is established when graphene serves as nanocircuit elements. We demonstrate that these two modulation effects can be dynamically controlled by appropriately integrating graphene with plasmonic nanostructures, which provide an active window for efficient modulation of surface plasmons. Our findings may pave the way towards realizing dynamic control of plasmonic response, which holds great potential applications in graphene-based active nanoplasmonic devices.