We use Hubble Space Telescope (HST) and ground-based imaging to study the multiple populations of 47 Tucanae (47 Tuc), combining high-precision photometry with calculations of synthetic spectra. Using filters covering a wide range of wavelengths, our HST photometry splits the main sequence into two branches, and we find that this duality is repeated in the subgiant and red giant regions, and on the horizontal branch. We calculate theoretical stellar atmospheres for main-sequence stars, assuming different chemical composition mixtures, and we compare their predicted colors through the HST filters with our observed colors. We find that we can match the complex of observed colors with a pair of populations, one with primeval abundance and another with enhanced nitrogen and a small helium enhancement, but with depleted C and O. We confirm that models of red giant and red horizontal branch stars with that pair of compositions also give colors that fit our observations. We suggest that the different strengths of molecular bands of OH, CN, CH, and NH, falling in different photometric bands, are responsible for the color splits of the two populations. Near the cluster center, in each portion of the color-magnitude diagram the population with primeval abundances makes up only ~20% of the stars, a fraction that increases outward, approaching equality in the outskirts of the cluster, with a fraction ~30% averaged over the whole cluster. Thus the second, He/N-enriched population is more concentrated and contributes the majority of the present-day stellar content of the cluster. We present evidence that the color-magnitude diagram of 47 Tuc consists of intertwined sequences of the two populations, whose separate identities can be followed continuously from the main sequence up to the red giant branch, and thence to the horizontal branch. A third population is visible only in the subgiant branch, where it includes ~8% of the stars. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.