To date, scanning tunneling microscopy (STM) has significantly advanced our understanding of the local structure, electronic properties, and reactivity of semiconductor surfaces. Less recognized is the fact that STM can also elucidate the local structural and electronic properties of low-dimensional materials. Herein, we demonstrate that STM can provide new and essential insight into the physical complexities of high-temperature copper oxide superconductors that is unavailable from conventional studies. After reviewing the basic theoretical concepts needed to evaluate STM data, we discuss several specific cases that illustrate the unique information that STM provides. These examples include (1) the atomic level nature of structural disorder in the BiO and TlO layers of Bi2Sr2CaCu2O8 and Tl2Ba2CaCu2O8 and the low-energy electronic states associated with these structural features, (2) the local structure and electronic consequences of metal substitution and oxygen doping in Bi2Sr2CaCu2O8, and (3) low-temperature tunneling spectroscopy measurements of the superconducting energy gap. Lastly, we summarize exciting future avenues to pursue in STM studies of these and other materials.