In this paper, a dual-band metamaterial absorber in the terahertz frequencies is proposed and its refractive index sensing characteristics is analyzed. The metamaterial structure is designed using a square metal ring with four T-shaped strips loaded outside of the ring, where the metal periodic array is on top of a silicon wafer backed with a metal ground plane. The resonant frequencies of the absorber are at 0.89 and 1.36 THz, whose absorption rates are both over 99% under normal TE and TM polarized incidences. The full widths at half maximum of them are 4.4 and 11.2 GHz, respectively, resulting in high quality factors (Q-factors) for these two frequency bands. The absorption rate of the absorber remains stable as the incident and polarized angles are changed. Several proposed metamaterial absorbers are experimentally fabricated and electron beam lithography (EBL) technology is employed. Good measurement results of the dual-band absorption performance are obtained using a terahertz time-domain spectroscopy system based on photoconductive antennas. Furthermore, the metamaterial absorber also shows sensing properties for analytes with different refractive indices or thicknesses. This work provides a new choice for the design of high-Q dual-band terahertz metamaterial absorbers and their application to refractive index sensing.