We present the Atacama Large Millimeter/submillimeter Array survey of CO(2-1) emission from the 1/5 solar metallicity, Local Group dwarf galaxy NGC 6822. We achieve high (0.9"~2pc) spatial resolution while covering a large area: four 250pcx250pc regions that encompass ~2/3 of NGC 6822's star formation. In these regions, we resolve ~150 compact CO clumps that have small radii (~2-3pc), narrow line width (~1km/s), and low filling factor across the galaxy. This is consistent with other recent studies of low-metallicity galaxies, but here shown with a 15x larger sample. At parsec scales, CO emission correlates with 8{mu}m emission better than with 24{mu}m emission and anticorrelates with H{alpha}, so that polycyclic aromatic hydrocarbon emission may be an effective tracer of molecular gas at low metallicity. The properties of the CO clumps resemble those of similar-size structures in Galactic clouds except of slightly lower surface brightness and with CO-to-H_2_ ratio ~1-2x the Galactic value. The clumps exist inside larger atomic-molecular complexes with masses typical for giant molecular clouds. Using dust to trace H_2_ for the entire complex, we find the CO-to-H_2_ ratio to be ~20-25x the Galactic value, but with strong dependence on spatial scale and variations between complexes that may track their evolutionary state. The H_2_-to-HI ratio is low globally and only mildly above unity within the complexes. The ratio of star formation rate to H_2_ is ~3-5x higher in the complexes than in massive disk galaxies, but after accounting for the bias from targeting star-forming regions, we conclude that the global molecular gas depletion time may be as long as in massive disk galaxies.