We present the discovery of the X-ray and optical afterglows of the short-duration GRB 150101B, pinpointing the event to an early-type host galaxy at z = 0.1343 ± 0.0030. This makes GRB 150101B the most nearby short GRB with an early-type host galaxy discovered to date. Fitting the spectral energy distribution of the host galaxy results in an inferred stellar mass of ≈ 7 × 10 10 M , stellar population age of ≈ 2 − 2.5 Gyr, and star formation rate of 0.4 M yr −1 . The host of GRB 150101B is one of the largest and most luminous short GRB host galaxies, with a B-band luminosity of ≈ 4.3L* and half-light radius of ≈ 8 kpc. GRB 150101B is located at a projected distance of 7.35 ± 0.07 kpc from its host center, and lies on a faint region of its host rest-frame optical light. Its location, combined with the lack of associated supernova, is consistent with a NS-NS/NS-BH merger progenitor. From modeling the evolution of the broad-band afterglow, we calculate isotropic-equivalent gamma-ray and kinetic energies of ≈ 1.3 × 10 49 erg and ≈ (6 − 14) × 10 51 erg, respectively, a circumburst density of ≈ (0.8 − 4) × 10 −5 cm −3 , and a jet opening angle of 9• . Using observations extending to ≈ 30 days, we place upper limits of (2 − 4) × 10 41 erg s −1 on associated kilonova emission. We compare searches following previous short GRBs to existing kilonova models, and demonstrate the difficulty of performing effective kilonova searches from cosmological short GRBs using current ground-based facilities. We show that at the Advanced LIGO/VIRGO horizon distance of 200 Mpc, searches reaching depths of ≈ 23 − 24 AB mag are necessary to probe a meaningful range of kilonova models.