We present new deep VI ground-based photometry of the Local Group dwarf galaxy Phoenix. Our results confirm that this galaxy is mainly dominated by red stars, with some blue plume stars indicating recent (100 Myr old) star formation in the central part of the galaxy. We have performed an analysis of the structural parameters of Phoenix based on an ESO/SRC scanned plate, in order to search for differentiated components. The elliptical isopleths show a sharp rotation of ~=90 deg of their major axis at radius r~=115^'' from the center, suggesting the existence of two components: an inner component facing in the east-west direction, which contains all the young stars, and an outer component oriented north-south, which seems to be predominantly populated by old stars. These results were then used to obtain the color-magnitude diagrams for three different regions of Phoenix in order to study the variation of the properties of its stellar population. The young population located in the central component of Phoenix shows a clear asymmetry in its distribution, with the younger blue plume stars predominantly located in the western half of the central component and the older core helium-burning stars predominantly situated in the east. This spatial variation could indicate a propagation of star formation across the central component. The H I cloud found at ~6' southwest by Young & Lo could have been involved in this process. We also find a decreasing gradient in the density of the intermediate-age population with the galactocentric radius, based on the number of stars populating the red clump in the color-magnitude diagram. Since no metallicity gradient is apparent, this indicates the presence of a substantial intermediate-age population in the central region of Phoenix that would be less abundant or absent in its outer regions. This result is also consistent with the gradient found in the number of horizontal-branch stars, whose frequency relative to red giant branch stars increases toward the outer part of the galaxy. These results, together with those of our morphological study, suggest the existence of an old, metal-poor population with a spheroidal distribution surrounding the younger inner component of Phoenix. This two-component structure may resemble the halo-disk structure observed in spirals, although more data, in particular on kinematics, are necessary to confirm this. We have estimated the average star formation rate for the last 1 Gyr and for the age interval 1-15 Gyr from the number of blue and red giant branch and asymptotic giant branch stars observed in the color-magnitude diagram. For the central region, the average past star formation rate is very similar to that for the last 1 Gyr. The recent star formation rate of Phoenix is also comparable to that displayed by typical dIrr galaxies, except perhaps for the fact that it lacks any strong very recent burst as exhibited by galaxies such as Sextans A or NGC 6822. The area-normalized star formation rate for the central region of Phoenix is in the range obtained by Hunter & Gallagher for their sample of dIrr galaxies. We have determined a distance modulus for Phoenix of (m-M)_0=23.0+/-0.1 using the tip of the red giant branch as a distance indicator. We find four short-period variable candidates from our photometry that might be anomalous Cepheids or W Vir stars. Finally, it is very unlikely that Phoenix has globular clusters, as is expected for a galaxy with such a faint absolute magnitude.