  We report a comprehensive experimental investigation on the magnetic
anisotropy in bulk single crystals of Cr$_2$Ge$_2$Te$_6$, a
quasi-two-dimensional ferromagnet belonging to the family of magnetic layered
transition metal trichalcogenides that have attracted recently a big deal of
interest with regard to the fundamental and applied aspects of two-dimensional
magnetism. For this purpose electron spin resonance (ESR) and ferromagnetic
resonance (FMR) measurements have been carried out over a wide frequency and
temperature range. A gradual change in the angular dependence of the ESR
linewidth at temperatures above the ferromagnetic transition temperature
$T_{\rm c}$ reveals the development of two-dimensional spin correlations in the
vicinity of $T_{\rm c}$ thereby proving the intrinsically low-dimensional
character of spin dynamics in Cr$_2$Ge$_2$Te$_6$. Angular and frequency
dependent measurements in the ferromagnetic phase clearly show an easy-axis
type anisotropy of this compound. Furthermore, these experiments are compared
with simulations based on a phenomenological approach, which takes into account
results of static magnetization measurements as well as high temperature $g$
factors obtained from ESR spectroscopy in the paramagnetic phase. As a result
the determined magnetocrystalline anisotropy energy density (MAE) $K_U$ is
$(0.48 \pm 0.02) \times10^6$ erg/cm$^3$. This analysis is complemented by
density functional calculations which yield the experimental MAE value for a
particular value of the electronic correlation strength $U$. The analysis of
the electronic structure reveals that the low-lying conduction band carries
almost completely spin-polarized, quasi-homogeneous, two-dimensional states.
