Since the early stages of operation of ground-based gravitational-wave interferometers, careful monitoring of these detectors has been an important component of their successful operation and observations. Characterization of gravitational-wave detectors blends computational and instrumental methods of investigating the detector performance. These efforts focus both on identifying ways to improve detector sensitivity for future observations and understand the non-idealized features in data that has already been recorded. Alongside a focus on the detectors themselves, detector characterization includes careful studies of how astrophysical analyses are affected by different data quality issues. This article presents an overview of the multifaceted aspects of the characterization of interferometric gravitational-wave detectors, including investigations of instrumental performance, characterization of interferometer data quality, and the identification and mitigation of data quality issues that impact analysis of gravitational-wave events. Looking forward, we discuss efforts to adapt current detector characterization methods to meet the changing needs of gravitational-wave astronomy.