  Magnetic anisotropy is crucially important for the stabilization of
two-dimensional (2D) magnetism, which is rare in nature but highly desirable in
spintronics and for advancing fundamental knowledge. Recent works on CrI$_3$
and CrGeTe$_3$ monolayers not only led to observations of the long-time-sought
2D ferromagnetism, but also revealed distinct magnetic anisotropy in the two
systems, namely Ising behavior for CrI$_3$ versus Heisenberg behavior for
CrGeTe$_3$. Such magnetic difference strongly contrasts with structural and
electronic similarities of these two materials, and understanding it at a
microscopic scale should be of large benefits. Here, first-principles
calculations are performed and analyzed to develop a simple Hamiltonian, to
investigate magnetic anisotropy of CrI$_3$ and CrGeTe$_3$ monolayers. The
anisotropic exchange coupling in both systems is surprisingly determined to be
of Kitaev-type. Moreover, the interplay between this Kitaev interaction and
single ion anisotropy (SIA) is found to naturally explain the different
magnetic behaviors of CrI$_3$ and CrGeTe$_3$. Finally, both the Kitaev
interaction and SIA are further found to be induced by spin-orbit coupling of
the heavy ligands (I of CrI$_3$ or Te of CrGeTe$_3$) rather than the commonly
believed 3d magnetic Cr ions.
