We present the detailed science case, and brief descriptions of the telescope design, site, and first light instrument plans for a new ultra-wide field submillimeter observatory, CCAT-prime, that we are constructing at a 5600 m elevation site on Cerro Chajnantor in northern Chile. Our science goals are to study star and galaxy formation from the epoch of reionization to the present, investigate the growth of structure in the Universe, improve the precision of B-mode CMB measurements, and investigate the interstellar medium and star formation in the Galaxy and nearby galaxies through spectroscopic, polarimetric, and broadband surveys at wavelengths from 200 m to 2 mm. These goals are realized with our two first light instruments, a large field-of-view (FoV) bolometer-based imager called Prime-Cam (that has both camera and an imaging spectrometer modules), and a multi-beam submillimeter heterodyne spectrometer, CHAI. CCAT-prime will have very high surface accuracy and very low system emissivity, so that combined with its wide FoV at the unsurpassed CCAT site our telescope/instrumentation combination is ideally suited to pursue this science. The CCAT-prime telescope is being designed and built by Vertex Antennentechnik GmbH. We expect to achieve first light in the spring of 2021.