New Hubble Space Telescope WFPC2 imaging of the dwarf starburst galaxy NGC 625 is presented. These data, which are 80% complete to V and I magnitudes of 26.0 and 25.3, respectively, allow us to study the recent star formation history of NGC 625. Using outlying red giant stars, we derive a tip of the red giant branch (TRGB) distance modulus of 27.95 ± 0.07. This corresponds to a distance of 3.89 ± 0.22 Mpc, placing this system on the far side of the Sculptor Group. NGC 625 has a well-defined radial stellar population gradient, evidenced by a central concentration of young main-sequence stars and a red giant branch (RGB)/asymptotic giant branch (AGB) ratio that increases with galactocentric distance. The prominent AGB is very red, similar to the population found in the Local Group dwarf irregular galaxy NGC 6822. The RGB stars can be detected out far from the central star-forming activity and show an elliptical distribution in agreement with the galaxy's outer H I distribution. Using Hα and Hβ narrowband imaging and previous optical spectroscopy, we identify substantial and varying internal extinction associated with the central active star formation regions. This extinction, which varies from A_V = 0.0 to 0.6 mag, hampers efforts to derive a detailed recent star formation history. To better understand the effects of internal extinction on the analysis of young stellar populations, synthetic models are presented that, for the first time, examine and account for this effect. Using the luminous blue helium-burning stars, we construct a simple model of the recent (<100 Myr) star formation in which an elevated but declining star formation rate has been present over this entire period. This is at odds with the presence of spectroscopic Wolf-Rayet (W-R) features in the major star formation region, which implies a short duration (≤5 Myr) for the recent starburst. This suggests that starbursts displaying W-R features are not necessarily all of a short duration. Finally, we speculate on the possible causes of the present burst of star formation in this apparently isolated galaxy and compare it with other nearby, well-studied dwarf starburst systems.