We compile a new sample of 115 nearby, non-Seyfert galaxies spanning a wide range of star formation activities, from starburst to nearly dormant, based on ultraviolet observations with various satellites. We combine these observations with infrared observations to study the relation between ratio of total far-infrared to ultraviolet luminosity and ultraviolet spectral slope (the IRX-UV relation). We show that, at fixed ultraviolet spectral slope, quiescent star-forming galaxies in our sample have systematically lower ratio of total far-infrared to ultraviolet luminosity than starburst galaxies. The strengths of spectral indices sensitive to star formation history, such as the 4000-Å spectral discontinuity and the Hα emission equivalent width, correlate well with distance from the mean relation for starburst galaxies in the IRX-UV diagram, while there is little or no correlation between the dust-sensitive Hα/Hβ ratio and this distance. This is strong observational evidence that the star formation history is relevant to the second parameter affecting the IRX-UV relation. We show that these results can be understood in the framework of the simple model of Charlot & Fall for the transfer of starlight through the interstellar medium in galaxies. We confirm that, for starburst galaxies, the tight IRX-UV relation can be understood most simply as a sequence in overall dust content. In addition, we find that the broadening of the relation for quiescent star-forming galaxies can be understood most simply as a sequence in the ratio of present to past-averaged star formation rate. We use a library of Monte Carlo realizations of galaxies with different star formation histories and dust contents to quantify the accuracy to which the ultraviolet attenuation A_FUV of a galaxy can be estimated from either the ratio of far-infrared to ultraviolet luminosity or the ultraviolet spectral slope. We provide simple formulae for estimating A_FUV as a function of either of these observational quantities and show that the accuracy of these estimates can be significantly improved if some constraints are available on the ratio of present to past-averaged star formation rate.