  Using a simple two-color selection based on $g$-, $z$-, and $K$-band
photometry, we pick out 1609 star-forming galaxies (sgzKs) and 422 passively
evolving galaxies (pgzKs) at z\sim2$ from a $K$-band-selected sample ($K_{\rm
AB} < 22.0$) in an area of $\sim 0.44$ deg$^{2}$ of the All-wavelength Extended
Groth Strip International Survey. The number counts of pgzKs\ in our sample
turn over at $K_{\rm AB} \sim 21.0$, and both the number of faint and bright
objects (including sgzKs\ and pgzKs) exceed the predictions of a recent
semi-analytic model of galaxy formation, a more successful model is need to
explain this diversity. We also find that the star formation rate (SFR) and
specific SFR (sSFR) of sgzKs\ increases with redshift at all masses, implying
that star-forming galaxies were much more active on average in the past.
Moreover, the sSFR of massive galaxies is lower at all redshifts, suggesting
that star formation contributes more to the mass growth of low-mass galaxies
than to high-mass galaxies. From {\it Hubble Space Telescope} Wide Field Camera
3 near-infrared imaging data, we find that morphologies of $z\sim2$ galaxies
not only have diffuse structures with lower $G$ and higher $M_{20}$ values, but
also have single-object morphologies (higher $G$ and lower $M_{20}$), implying
that there are morphological variety and different formation process for these
galaxies at $z\sim2$. Finally, we also study the fraction of active galactic
nuclei (AGNs) in the gzKs, 82 of 828 gzKs\ with four IRAC bands can be
classified as AGNs ($\sim$ 10%). Most of these AGN candidates have $L_{\rm
0.5-10\ keV}>10^{41}\,\rm erg\,s^{-1}$.
