This paper presents a dynamically tunable narrow/broad band switchable THz absorber. The absorber is composed of tunable materials, i.e., graphene and VO₂, and gold and cycloolefin copolymers. When VO₂ is thermally induced to achieve insulator-to-metal transition, the absorber switches between broad bands and narrow bands. Its switching amplitude can reach 98.9%. When VO₂ is in metallic phase, the absorber has a wide absorption bandwidth. At that time, the bandwidth of the absorber can be adjusted by changing the Fermi level of graphene. The absorber also has characteristic of wide-angle absorption for transverse electric (TE) and transverse magnetic (TM) waves. When VO₂ is in insulating state, the device switches to a narrowband absorber with a sensitivity of 439 GHz/RIU, which can be used in sensor. The THz absorber has many potential applications in multi-functional devices such as modulation, sensing, and electromagnetic cloaking.