We present a quantitative analysis of the star formation history (SFH) of 12 fields in the Small Magellanic Cloud (SMC) based on unprecedented deep [(B - R), R] color-magnitude diagrams (CMDs). Our fields reach down to the oldest main-sequence turnoff with a high photometric accuracy, which is vital for obtaining accurate SFHs, particularly at intermediate and old ages. We use the IAC-pop code to obtain the SFH, using synthetic CMDs generated with IAC-star. We obtain the SFH as a function ψ(t, z) of age and metallicity. We also consider several auxiliary functions: the initial mass function (IMF), phi(m), and a function accounting for the frequency and relative mass distribution of binary stars, β(f, q). We find that there are several main periods of enhancement of star formation: a young one peaked at ~0.2-0.5 Gyr old, only present in the eastern and in the central-most fields; two at intermediate ages present in all fields: a conspicuous one peaked at ~4-5 Gyr, and a less significant one peaked at ~1.5-2.5; and an old one, peaked at ~10 Gyr in all fields but the western ones. In the western fields, this old enhancement splits into two, one peaked at ~8 Gyr old and another at ~12 Gyr old. This "two-enhancement" zone is unaffected by our choice of stellar evolutionary library but more data covering other fields of the SMC are necessary in order to ascertain its significancy. Correlation between star formation rate enhancements and SMC-Milky Way encounters is not clear. Some correlation could exist with encounters taken from the orbit determination of Kallivayalil et al. But our results would also fit in a first pericenter passage scenario like the one claimed by Besla et al. For SMC-Large Magellanic Cloud encounters, we find a correlation only for the most recent encounter ~0.2 Gyr ago. This coincides with the youngest ψ(t) enhancement peaked at these ages in our eastern fields. The population younger than 1 Gyr represents ~7%-12% of the total ψ(t) in our fields of the wing area. This is not an exceptional increment as compared with the average ψ(t) but is very significant in the sense that these eastern fields are the only ones of this study in which star formation is currently going on. There is also a strong dichotomy between east/southeast and west in the current irregular shape of the SMC. We find that this dichotomy is produced by the youngest population and began ~1.0 Gyr ago or later. The age of the old population is similar at all radii and at all azimuth, and we constrain the age of this oldest population to be more than ~12 Gyr. We do not find yet a region dominated by a true, old, Milky-Way-like, halo at 4.5 kpc from the SMC center, indicating either that this old stellar halo does not exist in the SMC or that its contribution to the stellar populations, at the galactocentric distances of our outermost field, is negligible. Finally, we derive the age-metallicity relation and find that, in all fields, the metallicity increased continuously from early epochs until the present. This is in good agreement with the results from the Ca II triplet, a completely independent method, constituting an external consistency proof of IAC-pop in determining the chemical enrichment law.