  In this paper, we establish various maximal principles and develop the direct
moving planes and sliding methods for equations involving the physically
interesting (nonlocal) pseudo-relativistic Schr\"{o}dinger operators
$(-\Delta+m^{2})^{s}$ with $s\in(0,1)$ and mass $m>0$. As a consequence, we
also derive multiple applications of these direct methods. For instance, we
prove monotonicity, symmetry and uniqueness results for solutions to various
equations involving the operators $(-\Delta+m^{2})^{s}$ in bounded domains,
epigraph or $\mathbb{R}^{N}$, including pseudo-relativistic Schr\"odinger
equations, 3D boson star equations and the equations with De Giorgi type
nonlinearities.
