  The critical properties and magnetocaloric effect of semiconducting
ferrimagnet Mn$_3$Si$_2$Te$_6$ single crystals have been investigated by bulk
magnetization and heat capacity around $T_c$. Critical exponents $\beta =
0.41\pm0.01$ with a critical temperature $T_c = 74.18\pm0.08$ K and $\gamma =
1.21\pm0.02$ with $T_c = 74.35\pm0.05$ K are deduced by the Kouvel-Fisher plot,
whereas $\delta = 4.29\pm0.05(3.40\pm0.02)$ is obtained by a critical isotherm
analysis at $T = 74(75)$ K. The magnetic exchange distance is found to decay as
$J(r)\approx r^{-4.79}$, which lies between the mean-field and 3D Heisenberg
models. Moreover, the magnetic entropy change $-\Delta S_M$ features a maximum
at $T_c$, i.e., $-\Delta S_M^{max} \sim$ 2.53(1.67) J kg$^{-1}$ K$^{-1}$ with
in-plane(out-of-plane) field change of 5 T, confirming large magnetic
anisotropy. The heat capacity measurement further gives $-\Delta S_M^{max}$
$\sim$ 2.94 J kg$^{-1}$ K$^{-1}$ and the corresponding adiabatic temperature
change $\Delta T_{ad}$ $\sim$ 1.14 K with out-of-plane field change of 9 T.
