Gamma-ray bursts (GRBs) are relativistic explosions which originate at cosmological distances, and are among the most luminous transients in the universe. Following the prompt gamma-ray emission, a fading synchrotron "afterglow" is detectable at lower energies. While long-duration GRBs (duration > 2 sec) are linked to the deaths of massive stars, the progenitors of short-duration GRBs (duration < 2 sec) have remained elusive. Theoretical predictions formulated over the past two decades have suggested that they are the mergers of two compact objects, involving either two neutron stars (NS-NS) or a neutron star and a black hole (NS-BH). Such merging systems are also important to understand because they are premier candidates for gravitational wave detections with upcoming facilities and are considered likely sites of heavy element nucleosynthesis. The launch of the Swift satellite in 2004, with its rapid multi-wavelength monitoring and localization capabilities, led to the first discoveries of short GRB afterglows and therefore robust associations to host galaxies. At a Swift detection rate of ~8 events per year, the growing number of well-localized short GRBs enables comprehensive population studies of their afterglows and environments for the first time. In this thesis, I undertake a multi-wavelength observational campaign to address testable predictions for the progenitors of short GRBs. From their local environments, I show that short GRBs explode in diffuse regions of their host galaxies and are weakly correlated with the distribution of stellar mass and star formation in their host galaxies. I study the host galaxy demographics for the entire population and find that ~20-40% of short GRBs originate from elliptical galaxies, implying an older stellar progenitor. From their afterglows, I present evidence that some short GRBs are collimated in narrow jets of ~5-10 degrees, directly affecting the true energy scale and event rate. Finally, taking advantage of a decade of broad-band afterglow observations at radio through X-ray wavelengths, I find that short GRBs have median isotropic-equivalent energies of ~10^51 erg and that their local environments have low densities, ~10^-3-10^-2 cm^-3. Taken together, this thesis comprises several lines of independent evidence to demonstrate that short GRBs originate from the mergers of two compact objects, and also provides the first constraints on the explosion properties for a large sample of events. With the direct detection of gravitational waves from compact object mergers on the horizon, these studies provide necessary inputs to inform the next decade of joint electromagnetic-gravitational wave search strategies. / Astronomy