The abundances of iron and oxygen are homogeneously determined in a sample of 523 nearby (d<150 pc) FGK disk and halo stars with metallicities in the range -1.5<[Fe/H]<0.5. Iron abundances were obtained from an LTE analysis of a large set of Fe I and Fe II lines with reliable atomic data. Oxygen abundances were inferred from a restricted non-LTE analysis of the 777 nm O I triplet. We adopted the infrared flux method temperature scale and surface gravities based on Hipparcos trigonometric parallaxes. Within this framework, the ionization balance of iron lines is not satisfied: the mean abundances from the Fe I lines are systematically lower by 0.06 dex than those from the Fe II lines for dwarf stars of Teff>5500 K and [Fe/H]<0.0, and giant stars of all temperatures and metallicities covered by our sample. The discrepancy worsens for cooler and metal-rich main-sequence stars. We use the stellar kinematics to compute the probabilities of our sample stars to be members of the thin disk, thick disk, or halo of the Galaxy. We find that the majority of the kinematically-selected thick-disk stars show larger [O/Fe] ratios compared to thin-disk stars while the rest show thin-disk abundances, which suggests that the latter are thin-disk members with unusual (hotter) kinematics. A close examination of this pattern for disk stars with ambiguous probabilities shows that an intermediate population with properties between those of the thin and thick disks does not exist, at least in the solar neighborhood. Excluding the stars with unusual kinematics, we find that thick-disk stars show slowly decreasing [O/Fe] ratios from about 0.5 to 0.4 in the -0.8<[Fe/H]<-0.3 range. Using a simple model for the chemical evolution of the thick disk we show that this trend results directly from the metallicity dependence of the Type II supernova yields. At [Fe/H]>-0.3, we find no obvious indication of a sudden decrease (i.e., a "knee") in the [O/Fe] vs. [Fe/H] pattern of thick-disk stars that would connect the thick and thin disk trends at a high metallicity. We conclude that Type Ia supernovae (SN Ia) did not contribute significantly to the chemical enrichment of the thick disk. In the -0.8<[Fe/H]<+0.3 range, thin-disk stars show decreasing [O/Fe] ratios from about 0.4 to 0.0 that require a SN Ia contribution. The implications of these results for studies of the formation and evolution of the Galactic disk are discussed. Tables 4-6 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/465/271 Partially based on observations obtained with the Hobby-Eberly Telescope, which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen; and data from the UVES Paranal Observatory Project (ESO DDT Program ID 266.D-5655).