Approximately 20-30% of 1.4<z<2.5 galaxies with K<22 (Vega) detected with Spitzer MIPS at 24um 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 a spectral energy distribution peaking longward of 1.6um 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 Active Galactic Nuclei (AGNs) with column densities N_H~10^{24} cm^-2 and an average, unobscured X-ray luminosity L_{2-8 keV}~(1-4)x10^43 erg/s. Their sky density(~3200 deg^-2) and space density (~2.6x10^-4 Mpc^-3) are twice those of X-ray detected AGNs at z~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 ~50-60% for M~10^11 M_sun, an effect likely connected with downsizing in galaxy formation. The ratio of theinferred 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 inlocal spheroids, implying concurrent growth of both within the precursors oftodays massive galaxies.