The highly redshifted, sky-averaged spectrum from neutral hydrogen's 21-cm line is a promising probe of the history of the universe after the release of the Cosmic Microwave background and prior to the reionization of the universe by the first generations of compact sources. It may reveal when the first stars and black holes formed, when they began to effectively heat the hydrogen pervading the universe, and by what time they had reionized the universe. It could also reveal exotic physics, such as non-gravitational interactions between dark matter and hydrogen or dark matter self-annihilation. However, to measure this global signal, one must extract it from very large galactic foregrounds, noise, and instrumental effects. This thesis lays out a software pipeline designed both to perform this extraction and to aid in optimization of experimental design. In addition, it examines in detail the claimed detection of a 600 mK absorption trough centered at 78 MHz in the sky averaged radio spectrum from the Experiment to Detect the Global Epoch of Reionization Signal (EDGES). While the community has proposed many models to explain why the signal is deeper than previously thought possible, a relative few have advanced the possibility that the signal is artificial or a product of inaccurate modeling, as will be explored here. The work presented here illustrates the great potential of 21-cm cosmology to unlock some of the most pressing unknowns in our current knowledge of the universe.