This paper proposes a terahertz absorber with a simple four-layered structure that can be dynamically switched between broadband and triple-band by controlling the chemical potential of graphene. The proposed absorber owns broadband absorption in the frequency range from 5.28 THz to 7.86 THz with the corresponding absorption efficiency above 90%, when the chemical potential of graphene is 150 meV. By increasing the chemical potential of graphene to 550 meV, the broadband absorption splits into triple-band absorption, with the peak locating at 5.39 THz, 7.01 THz and 8.1 THz, respectively. Detailed investigation shows that the broadband absorption should originate from magnetic resonance, Fabry-Pérot cavity resonance and surface plasmon polariton. The triple-band absorption should arise from the combination of Fabry-Pérot cavity resonance and surface plasmon polariton. Additionally, both broadband absorption and triple-band absorption are insensitive to the incident polarization. This tunable and bifunctional metamaterial structure shows a great potential in terahertz applications, such as detectors, modulators and sensors.