We present multiwavelength observations of the afterglow of GRB 130427A, the brightest (in total fluence) gamma-ray burst (GRB) of the past 29 yr. Optical spectroscopy from Gemini-North reveals the redshift of the GRB to be z = 0.340, indicating that its unprecedented brightness is primarily the result of its relatively close proximity to Earth; the intrinsic luminosities of both the GRB and its afterglow are not extreme in comparison to other bright GRBs. We present a large suite of multiwavelength observations spanning from 300 s to 130 days after the burst and demonstrate that the afterglow shows relatively simple, smooth evolution at all frequencies, with no significant late-time flaring or rebrightening activity. The entire data set from 1 GHz to 10 GeV can be modeled as synchrotron emission from a combination of reverse and forward shocks in good agreement with the standard afterglow model, providing strong support to the applicability of the underlying theory and clarifying the nature of the GeV emission observed to last for minutes to hours following other very bright GRBs. A tenuous, wind-stratified circumburst density profile is required by the observations, suggesting a massive-star progenitor with a low mass-loss rate, perhaps due to low metallicity. GRBs similar in nature to GRB 130427A, inhabiting low-density media and exhibiting strong reverse shocks, are probably not uncommon but may have been difficult to recognize in the past owing to their relatively faint late-time radio emission; more such events should be found in abundance by the new generation of sensitive radio and millimeter instruments.