The Event Horizon Telescope (EHT) provides the unprecedented ability to directly resolve the structure and dynamics of black hole emission regions on scales smaller than their horizons. This has the potential to critically probe the mechanisms by which black holes accrete and launch outflows, and the structure of supermassive black hole spacetimes. However, accessing this information is a formidable analysis challenge for two reasons. First, the EHT natively produces a variety of data types that encode information about the image structure in nontrivial ways; these are subject to a variety of systematic effects associated with very long baseline interferometry and are supplemented by a wide variety of auxiliary data on the primary EHT targets from decades of other observations. Second, models of the emission regions and their interaction with the black hole are complex, highly uncertain, and computationally expensive to construct. As a result, the scientific utilization of EHT observations requires a flexible, extensible, and powerful analysis framework. We present such a framework, Themis, which defines a set of interfaces between models, data, and sampling algorithms that facilitates future development. We describe the design and currently existing components of Themis, how Themis has been validated thus far, and present additional analyses made possible by Themis that illustrate its capabilities. Importantly, we demonstrate that Themis is able to reproduce prior EHT analyses, extend these, and do so in a computationally efficient manner that can efficiently exploit modern high-performance computing facilities. Themis has already been used extensively in the scientific analysis and interpretation of the first EHT observations of M87.