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 gamma-ray burst (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 approximate to 7 x 10(10) M-circle dot, stellar population age of approximate to 2-2.5 Gyr, and star formation rate of less than or similar to 0.4M(circle dot) 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 approximate to 4.3L(*) and half-light radius of approximate to 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 an NS-NS/NS-BH merger progenitor. From modeling the evolution of the broadband afterglow, we calculate isotropic-equivalent gamma-ray and kinetic energies of approximate to 1.3 x 10(49) erg and approximate to(6-14) x 10(51) erg, respectively, a circumburst density of approximate to(0.8-4) x 10(-5) cm(-3), and a jet opening angle of greater than or similar to 9 degrees. Using observations extending to approximate to 30 days, we place upper limits of less than or similar to(2-4) x 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 approximate to 23-24 AB. mag are necessary to probe a meaningful range of kilonova models.