The He, C, N, and O abundances in more than 120 planetary nebulae (PNe) of our Galaxy and the Magellanic Clouds have been redetermined by analyzing new PNe observations. The characteristics of PNe obtained bymodeling their spectra have been used to compile a new catalog of parameters forGalactic and extragalactic PNe, which is accessible at http://www.astro.spbu.ru/staff/afk/GalChemEvol.html.The errors in the parameters of PNe and their elemental abundances related to inaccuracies in the observational data have been analyzed. The He abundance is determined with an accuracy of 0.06 dex, while the errors in the C, N, and O abundances are 0.1-0.2 dex. Taking into account the inaccuracies in the corrections for the ionization stages of the elements whose lines are absent in the PNe spectra increases the errors in the He abundance to 0.1 dex and in the C, N, and O abundances to 0.2-0.3 dex. The elemental abundances in PNe of various Galactic subsystems and the Magellanic Clouds have been analyzed. This analysis suggests that the Galactic bulge objects are similar to type II PNe in Peimbert's classification, whose progenitor stars belong to the thin-disk population with ages of at least 4-6 Gyr. A similarity between the elemental abundances in PNe of the Magellanic Clouds and the Galactic halo has been established.