We have obtained deep CCD photometry of approximately 16 000 stars in the Leo I dwarf spheroidal galaxy (dSph), the most distant known satellite of the Milky Way Galaxy. The resulting color-magnitude diagrams (CMDs) of Leo I reveal several striking features. (1) There is a well-defined red giant branch (RGB) which is very blue and has a flatter slope than all other dwarf dSphs and Galactic globular clusters. The tip of the RGB is at I(TRGB)=18.25+/-0.10 mag and (V-I)TRGB=1.434-0.02 mag. (2) Unlike all of the other known dSphs, Leo I has no obvious horizontal branch; however, it does show a strong clump of red giants at V almost-equal-to 22.4 mag and (V-I) almost-equal-to 0.8 mag. (3) The CMD of Leo I shows approximately 50 anomalous Cepheid candidates. (4) There are approximately 50 asymptotic giant branch stars above the tip of the RGB, including 15 known carbon stars. (5) A large number of stars is seen at V greater-than-or-similar-to 23 mag (M(V) greater-than-or-similar-to + 1 mag) and 0 < ( V-I) < 1 mag. The present data do not go sufficiently deep to determine conclusively the nature of these faint stars. However, despite increasing crowding effects, the number density of these stars increases from V=23 mag to V=24 mag, consistent with the presence of a young population of - 3 Gyr. From the I magnitude of the tip of the RGB we estimate the distance modulus of Leo I to be (m-M)0=22.18+/-0.11 mag, corresponding to a distance of 270+/-10 kpc for an adopted I-band extinction of 0.04 mag. We estimate the mean color of the RGB at M(I)=-3.5 mag to be (V-I)-3.5=1.29+/-0.02 mag with an intrinsic dispersion of +/-0.07 mag. These results suggest that the mean metallicity of the RGB stars is [Fe/H]=-2.0+/-0.1 dex with a dispersion of approximately 0.3 dex. Applying timing arguments to the pair of our Galaxy and Leo I, we obtain an estimate for the mass of our Galaxy of M = 1.7 X 10(12) M . for an assumed age of the Universe of t=14 Gyr.