We use archival Hubble Space Telescope observations of resolved stellar populations to derive the star formation histories (SFHs) of 18 nearby starburst dwarf galaxies. In this first paper, we present the observations, color-magnitude diagrams (CMDs), and the SFHs of the 18 starburst galaxies, based on a homogeneous approach to the data reduction, differential extinction, and treatment of photometric completeness. We adopt a star formation rate (SFR) threshold normalized to the average SFR of the individual system as a metric for classifying starbursts in SFHs derived from resolved stellar populations. This choice facilitates finding not only the currently bursting galaxies but also "fossil" bursts increasing the sample size of starburst galaxies in the nearby (D < 8 Mpc) universe. Thirteen of the eighteen galaxies are experiencing ongoing bursts and five galaxies show fossil bursts. From our reconstructed SFHs, it is evident that the elevated SFRs of a burst are sustained for hundreds of Myr with variations on small timescales. A long >100 Myr temporal baseline is thus fundamental to any starburst definition or identification method. The longer lived bursts rule out rapid "self-quenching" of starbursts on global scales. The bursting galaxies' gas consumption timescales are shorter than the Hubble time for all but one galaxy confirming the short-lived nature of starbursts based on fuel limitations. Additionally, we find that the strength of the Hα emission usually correlates with the CMD-based SFR during the last 4-10 Myr. However, in four cases, the Hα emission is significantly less than what is expected for models of starbursts; the discrepancy is due to the SFR changing on timescales of a few Myr. The inherently short timescale of the Hα emission limits identifying galaxies as starbursts based on the current characteristics which may or may not be representative of the recent SFH of a galaxy. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.