  The magnetism in Mn$_3$Si$_2$Te$_6$ has been investigated using thermodynamic
measurements, first principles calculations, neutron diffraction and diffuse
neutron scattering on single crystals. These data confirm that
Mn$_3$Si$_2$Te$_6$ is a ferrimagnet below a Curie temperature of $T_C$
approximately 78K. The magnetism is anisotropic, with magnetization and neutron
diffraction demonstrating that the moments lie within the basal plane of the
trigonal structure. The saturation magnetization of approximately 1.6$\mu_B$/Mn
at 5K originates from the different multiplicities of the two
antiferromagnetically-aligned Mn sites. First principles calculations reveal
antiferromagnetic exchange for the three nearest Mn-Mn pairs, which leads to a
competition between the ferrimagnetic ground state and three other magnetic
configurations. The ferrimagnetic state results from the energy associated with
the third-nearest neighbor interaction, and thus long-range interactions are
essential for the observed behavior. Diffuse magnetic scattering is observed
around the 002 Bragg reflection at 120K, which indicates the presence of strong
spin correlations well above $T_C$. These are promoted by the competing ground
states that result in a relative suppression of $T_C$, and may be associated
with a small ferromagnetic component that produces anisotropic magnetism below
$\approx$330K.
