  We report a detailed experimental and theoretical study on the effect of
hydrostatic pressure on the structural and magnetic aspects of the layered
honeycomb antiferromagent $\alpha$-RuCl$_{3}$. Magnetic susceptibility
measurements performed under almost ideal hydrostatic-pressure conditions yield
that the phase transition to zigzag-type antiferromagnetic order at $T_N$ = 7.3
K can be rapidly suppressed to about 6.1 K. A further suppression with
increasing pressure is impeded due to the occurrence of a pressure-induced
structural transition at $p \geq$ 104 MPa, accompanied by a strong dimerization
of Ru-Ru bonds, which gives rise to a collapse of the magnetic susceptibility.
Whereas the dimerization transition is strongly first order, as reflected by
large discontinuous changes in $\chi$ and pronounced hysteresis effects, the
magnetic transition under varying pressure and magnetic field also reveals
indications for a weakly first-order transition. We assign this observation to
a strong magnetoelastic coupling in this system. Measurements of $\chi$ under
varying pressure in the paramagnetic regime ($T > T_N$) and before dimerization
($p <$ 100 MPa) reveal a considerable increase of $\chi$ with pressure. These
experimental observations are consistent with the results of ab-initio Density
Functional Theory (DFT) calculations on the pressure-dependent structure and
the corresponding pressure-dependent magnetic model. Comparative susceptibility
measurements on a second crystal showing two consecutive magnetic transitions
instead of one, indicating the influence of stacking faults. Using different
temperature-pressure protocols the effect of these stacking faults can be
temporarily overcome, transforming the magnetic state from a multiple-$T_N$
into a single-$T_N$ state.
