We present optical photometry and spectra for the Type Ia supernova (SN Ia) 2007gi in the nearby galaxy NGC 4036. SN 2007gi is characterized by extremely high-velocity (HV) features of the intermediate-mass elements (Si, Ca, and S), with expansion velocities ($v_{\rm exp}$) approaching $\sim$15,500 km s$^{-1}$ near maximum brightness (compared to $\sim$10,600 km s$^{-1}$ for SNe Ia with normal $v_{\rm exp}$). SN 2007gi reached a $B$-band peak magnitude of 13.25$\pm$0.04 mag with a decline rate of $\Delta m_{15}(B)$(true) = 1.33$\pm$0.09 mag. The $B$-band light curve of SN 2007gi demonstrated an interesting two-stage evolution during the nebular phase, with a decay rate of 1.16$\pm$0.05 mag (100 days)$^{-1}$ during $t = 60$--90 days and 1.61$\pm0.04$ mag (100 days)$^{-1}$ thereafter. Such a behavior was also observed in the HV SN Ia 2006X, and might be caused by the interaction between supernova ejecta and circumstellar material (CSM) around HV SNe Ia. Based on a sample of a dozen well-observed $R$-band (or unfiltered) light curves of SNe Ia, we confirm that the HV events may have a faster rise time to maximum than the ones with normal $v_{\rm exp}$.Comment: 30 pages, 10 figure