As the endpoints of stars less massive than roughly eight solar masses, the population of Galactic white dwarfs (WD) contain information about complex stellar evolution processes. Associated pairs of WDs add an extra degree of leverage; both WDs must have formed and evolved together. The work presented in this dissertation uses various populations of double WDs (DWD) to constrain evolution of both single and binary stars. One example is the set of low-mass WDs with unseen WD companions, which are formed through a dynamically-unstable mass loss process called the common envelope. To work toward a quantitative understanding of the common envelope, we develop and apply a Bayesian statistical technique to identify the masses of the unseen WD companions. We provide results which can be compared to evolutionary models and hence a deeper understanding of how binary stars evolve through a common envelope. The statistical technique we develop can be applied to any population of single-line spectroscopic binaries.

Binaries widely separated enough that they avoid any significant interaction independently evolve into separate WDs that can be identified in photometric and astrometric surveys. We discuss techniques for finding these objects, known as wide DWDs. We present a catalog of 142 candidate wide DWDs, combining both previously detected systems and systems we identify in our searches in the Sloan Digital Sky Survey. Having been born at the same time, the masses and cooling ages of the WDs in wide DWDs, obtained with our spectroscopic follow-up campaign can be used to constrain the initial-final mass relation, which relates a main sequence star to the mass of the WD into which it will evolve. We develop a novel Bayesian technique to interpret our data and present our resulting constraints on this relation which are particularly strong for initial masses between two and four solar masses. During this process, we identified one wide DWD, HS 2220+2146, that was peculiar since the more massive WD in this system evolved second.

We construct an evolutionary formation scenario in which the system began as a hierarchical triple in which the inner binary merged (possibly due to Kozai-Lidov oscillations) forming a post-blue straggler binary. The system then evolved into the DWD we observe today. We further discuss the potential for identifying more wide DWDs, including peculiar systems like HS 2220+2146, in future surveys such as Gaia.