Spectra have been obtained with the Infrared Spectrograph (IRS) on the Spitzer Space Telescope for 20 sources in the Lockman Hole field of the SWIRE survey. The sample is divided between sources with indicators of an obscured AGN, based primarily on X-ray detections of optically faint sources, and sources with indicators of a starburst, based on optical and near-infrared spectral energy distributions (SEDs), which show a luminosity peak from stellar photospheric emission. Ten of the 11 AGN sources have IRS spectra that show silicate absorption or are power laws; only one AGN source shows PAH emission features. All nine of the sources showing starburst SEDs in the near-infrared show PAH emission features in the IRS spectra. Redshifts are determined from the IRS spectra for all nine starbursts (1.0 < z < 1.9) and 8 of the 11 AGNs (0.6 < z < 2.5). Classification as AGN because of an X-ray detection, the classification as AGN or starburst derived from the photometric SED, and the IRS spectroscopic classification as AGN (silicate absorption) or starburst (PAH emission) are all consistent in 18 of 20 sources. The surface density for starbursts that are most luminous in the mid-infrared is less than that for the most luminous AGNs within the redshift interval 1.7 z 1.9. This result implies that mid-infrared source counts at high redshift are dominated by AGNs for f_ν(24 μm) 1.0 mJy.