We analyze the association of galaxies to Lya and OVI absorption, the most commonly detected transitions in the low-z intergalactic medium (IGM), in the fields of 14 quasars with z_em = 0.06-0.57. Confirming previous studies, we observe a high covering fraction for Lya absorption to impact parameter rho = 300kpc: 33/37 of our L>0.01L* galaxies show Lya equivalent width W_Lya>50mA. Galaxies of all luminosity L>0.01L* and spectral type are surrounded by a diffuse and ionized circumgalactic medium (CGM), whose baryonic mass is estimated at ~10^(10.5 +/- 0.3) Msun for a constant N_H. The virialized halos and extended CGM of present-day galaxies are responsible for most strong Lya absorbers (W_Lya > 300mA) but cannot reproduce the majority of observed lines in the Lya forest. We conclude that the majority of Lya absorption with W_Lya=30-300mA occurs in the cosmic web predicted by cosmological simulations and estimate a characteristic width for these filaments of ~400kpc. Regarding OVI, we observe a near unity covering fraction to rho=200kpc for L>0.1L* galaxies and to rho = 300kpc for sub-L* (0.1 L*<L<L*) galaxies. Similar to our Lya results, stronger OVI systems (W_OVI > 70mA) arise in the virialized halos of L>0.1L* galaxies. Unlike Lya, the weaker OVI systems (W_OVI~30mA) arise in the extended CGM of sub-L* galaxies. The majority of OVI gas observed in the low-z IGM is associated with a diffuse medium surrounding individual galaxies with L~0.3L*, and rarely originates in the so-called warm-hot IGM (WHIM) predicted by cosmological simulations.
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