  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 ~7x10^10
M_sol, stellar population age of ~2-2.5 Gyr, and star formation rate of <0.4
M_sol 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 ~1.3x10^49 erg and ~(6-14)x10^51 erg,
respectively, a circumburst density of ~(0.8-4)x10^-5 cm^-3, and a jet opening
angle of >9 deg. Using observations extending to ~30 days, we place upper
limits of <(2-4)x10^41 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.
