We present a method for determining mean light-weighted ages and abundances of Fe, Mg, C, N, and Ca from medium-resolution spectroscopy of unresolved stellar populations. The method is implemented in a publicly available code called EZ_Ages. The method and error estimation are described, and the results tested for accuracy and consistency, by application to integrated spectra of well-known Galactic globular and open clusters. Ages and abundances from integrated light analysis agree with studies of resolved stars to within ±0.1 dex for most clusters, and to within ±0.2 dex for nearly all cases. The results are robust to the choice of Lick indices used in the fitting to within ±0.1 dex, except for a few systematic deviations that are clearly categorized. The realism of our error estimates is checked through comparison with detailed Monte Carlo simulations. Finally, we apply EZ_Ages to the sample of galaxies presented in Thomas et al. (2005) and compare our derived values of age, [Fe/H], and [α/Fe] to their analysis. We find that [α/Fe] is very consistent between the two analyses, that ages are consistent for old (age > 10 Gyr) populations but show modest systematic differences at younger ages, and that [Fe/H] is fairly consistent, with small systematic differences related to the age systematics. Overall, EZ_Ages provides accurate estimates of fundamental parameters from medium-resolution spectra of unresolved stellar populations in the old and intermediate-age regime, for the first time allowing quantitative estimates of the abundances of C, N, and Ca in these unresolved systems.