  We study galactic star-formation activity as a function of environment and
stellar mass over 0.5<z<2.0 using the FourStar Galaxy Evolution (ZFOURGE)
survey. We estimate the galaxy environment using a Bayesian-motivated measure
of the distance to the third nearest neighbor for galaxies to the stellar mass
completeness of our survey, $\log(M/M_\odot)>9 (9.5)$ at z=1.3 (2.0). This
method, when applied to a mock catalog with the photometric-redshift precision
($\sigma_z / (1+z) \lesssim 0.02$), recovers galaxies in low- and high-density
environments accurately. We quantify the environmental quenching efficiency,
and show that at z> 0.5 it depends on galaxy stellar mass, demonstrating that
the effects of quenching related to (stellar) mass and environment are not
separable. In high-density environments, the mass and environmental quenching
efficiencies are comparable for massive galaxies ($\log (M/M_\odot)\gtrsim$
10.5) at all redshifts. For lower mass galaxies ($\log (M/M)_\odot) \lesssim$
10), the environmental quenching efficiency is very low at $z\gtrsim$ 1.5, but
increases rapidly with decreasing redshift. Environmental quenching can account
for nearly all quiescent lower mass galaxies ($\log(M/M_\odot) \sim$ 9-10),
which appear primarily at $z\lesssim$ 1.0. The morphologies of lower mass
quiescent galaxies are inconsistent with those expected of recently quenched
star-forming galaxies. Some environmental process must transform the
morphologies on similar timescales as the environmental quenching itself. The
evolution of the environmental quenching favors models that combine gas
starvation (as galaxies become satellites) with gas exhaustion through
star-formation and outflows ("overconsumption"), and additional processes such
as galaxy interactions, tidal stripping and disk fading to account for the
morphological differences between the quiescent and star-forming galaxy
populations.
