The Nancy Grace Roman Space Telescope (“Roman”), under development by NASA, will investigate possible causes for the phenomenon of dark energy and detect and characterize extrasolar planets. The 2.4 m space telescope has two main instruments: a wide-field, infrared imager and a coronagraph. The coronagraph instrument (CGI) is a technology demonstrator designed to help bridge the gap between the current state-of-the-art space and ground instruments and future high-contrast space coronagraphs that will be capable of detecting and characterizing Earth-like planets in the habitable zones of other stars. Using adaptive optics, including two high-density deformable mirrors and low- and high-order wavefront sensing and control, CGI is designed to suppress the star light by up to nine orders of magnitude, potentially enabling the direct detection and characterization of Jupiter-class exoplanets. Contrast is the measure of starlight suppression, and high contrast is the chief virtue of a coronagraph. But it is not the only important characteristic: contrast must be balanced against acceptance of planet light. The remaining unsuppressed starlight must also have a stable morphology to allow further estimation and subtraction. To achieve all these goals in the presence of the disturbance and radiation environment of space, the coronagraph must be designed and fabricated as a highly optimized system. The CGI error budget is the top-level tool used to guide the optimization, enabling trades of various competing errors. The error budget is based on an analytical model, which enables rapid calculation and tracking of performance for the numerous and diverse questions that arise in the system engineering process. We outline the coronagraph system engineering approach and the error budget. We then describe in detail the analytical model for direct imaging and spectroscopy and show how it connects to the error budget. We introduce a number of useful ancillary metrics that provide insight into the capabilities of the instrument. Since models always need to be validated, we describe the validation approach for the CGI analytical model.