Approximately 20%-30% of 1.4 unk z unk 2.5 galaxies with unk < 22 detected with Spitzer MIPS at 24 mu m show excess mid-IR emission relative to that expected based on the rates of star formation measured from other multiwavelength data. These galaxies also display some near-IR excess in Spitzer IRAC data, with an SED peaking longward of 1.6 mu m in the rest frame, indicating the presence of warm dust emission usually absent in star-forming galaxies. Stacking Chandra data for the mid-IR excess galaxies yields a significant hard X-ray detection at rest-frame energies >6.2 keV. The stacked X-ray spectrum rises steeply at >10 keV, suggesting that these sources host Compton-thick AGNs with column densities N sub(H) unk 10 super(24) cm super(-2) and an average, unobscured X-ray luminosity unk approximately (1-4) x 10 super(43) ergs s super(-1). Their sky density ( similar to 3200 deg super(-2)) and space density ( similar to 2.6 x 10 super(-4) Mpc super(-3)) are twice those of X-ray-detected AGNs at z approximately 2, and much larger than those of previously known Compton-thick sources at similar redshifts. The mid-IR excess galaxies are part of the long sought after population of distant heavily obscured AGNs predicted by synthesis models of the X-ray background. The fraction of mid-IR excess objects increases with galaxy mass, reaching similar to 50%-60% for M similar to 10 super(11) M unk, an effect likely connected with downsizing in galaxy formation. The ratio of the inferred black hole growth rate from these Compton-thick sources to the global star formation rate at z = 2 is similar to the mass ratio of black holes to stars in local spheroids, implying concurrent growth of both within the precursors of today's massive galaxies.