Atacama Cosmology Telescope: Dusty star-forming galaxies and active galactic nuclei in the equatorial survey

We present a catalog of 510 radio-loud active galactic nuclei (AGN, primarily blazars) and 287 dusty star-forming galaxies (DSFGs) detected by the Atacama Cosmology Telescope at $>5\sigma$ significance in bands centered on 148 GHz (2 mm), 218 GHz (1.4 mm) and 277 GHz (1.1 mm), from a 480 square degrees strip on the celestial equator with additional (360 square degrees) shallower fields. Combining the deepest available 218 GHz wide-field imaging, 277 GHz data, and multi-band filtering yields the most sensitive wide-field millimeter-wave DSFG selection to date with rms noise referenced to 218 GHz reaching $<2$ mJy. We developed techniques to remove Galactic contamination from the extragalactic catalog, yielding 321 additional Galactic sources. We employ a new flux debiasing method that handles the heterogeneous sample selection due to Galactic cuts. We present spectral properties and source counts of the AGN and DSFGs. The DSFG spectra depart from an optically thin modified blackbody between 218 GHz and 277 GHz, consistent with optically thick emission or an additional cold dust component. For bright AGN, the inter-year RMS fractional deviation in flux density from source variability is $\sim40\%$. We report 8$-$2870 mJy source counts for AGN and 8$-$90 mJy source counts for DSFGs, the latter probing both the brighter, lensed population and the fainter, unlensed population. At 277 GHz we report the first source counts measurements at these flux densities, finding an excess above most model count predictions. Finally, we select thirty of the brightest DSFGs for multi-frequency study as candidate high-$z$ lensed systems.