Fano resonances in metamaterial are important due to their low-loss subradiant behavior that allows excitation of high quality (Q) factor resonances extending from the microwave to the optical bands. Fano resonances have recently showed their great potential in the areas of modulation, filtering, and sensing for their extremely narrow linewidths. However, the Fano resonances in a metamaterial system arise from the interaction of all that form the structure, limiting the tunability of the resonances. Besides, sensing trace analytes using Fano resonances are still challenging. In the present work, we demonstrate the excitation of Fano resonances in metamaterial consisting of a period array of two concentric double-split-ring resonators with symmetry breaking (position asymmetry and gaps asymmetry). The tunability and sensing of Fano resonances are both studied in detail. Introducing position asymmetry in the metamaterial leads to one Fano resonance located at 0.50 THz, while introducing gaps asymmetry results in two Fano resonances located at 0.35 THz and 0.50 THz. The transmittance, position, and linewidth of the three Fano resonances can be easily tuned by varying the asymmetry deviations. The Q factor and figure of merit (FoM) of Fano resonances with different asymmetry deviations are calculated for performance optimization. The Fano resonances having the highest FoM are used for the sensing of analytes at different refractive indices, and the Fano resonance performing the best in refractive index sensing is further applied to detect the analyte thickness. The results demonstrate that the tunable Fano resonances show tremendous potential in sensing applications, offering an approach to engineering highly efficient modulators and sensors.