The optical identifications are presented of the Einstein Extended Medium-Sensitivity Survey (EMSS), including the methodology used to optically identify the EMSS sources and the uncertainties involved with that process. The optical properties of the classes of X-ray, optical, and radio data for each of the identified and, as yet, unidentified sources of the survey are described. A new class of X-ray emitters, cooling flow galaxies, is proposed. The criteria used to determine whether the proposed optical counterpart to the X-ray source is a plausible identification are described. Plausibility is based on the optical classification of the counterpart, e.g., AGN, cluster, G star, and the X-ray-to-optical flux ratios previously observed for these classes of X-ray emitters. Two independent schemes of optical classification of the counterparts are used to check the plausibility of these identifications; one is based on moderate-resolution optical spectroscopy, and the other, on inferred X-ray luminosity and the overall energy distribution. 110 refs.