We present key results from the Herschel Orion Protostar Survey (HOPS): spectral energy distributions (SEDs) and model fits of 330 young stellar objects, predominantly protostars, in the Orion molecular clouds. This is the largest sample of protostars studied in a single, nearby star formation complex. With near-infrared photometry from 2MASS, mid- and far-infrared data from Spitzer and Herschel, and submillimeter photometry from APEX, our SEDs cover 1.2-870{mu}m and sample the peak of the protostellar envelope emission at ~100{mu}m. Using mid-IR spectral indices and bolometric temperatures, we classify our sample into 92 Class 0 protostars, 125 Class I protostars, 102 flat-spectrum sources, and 11 Class II pre-main-sequence stars. We implement a simple protostellar model (including a disk in an infalling envelope with outflow cavities) to generate a grid of 30400 model SEDs and use it to determine the best-fit model parameters for each protostar. We argue that far-IR data are essential for accurate constraints on protostellar envelope properties. We find that most protostars, and in particular the flat-spectrum sources, are well fit. The median envelope density and median inclination angle decrease from Class 0 to Class I to flat-spectrum protostars, despite the broad range in best-fit parameters in each of the three categories. We also discuss degeneracies in our model parameters. Our results confirm that the different protostellar classes generally correspond to an evolutionary sequence with a decreasing envelope infall rate, but the inclination angle also plays a role in the appearance, and thus interpretation, of the SEDs.