This dissertation describes a multidimensional relativistic hydrodynamic code which solves the special relativistic hydrodynamic equations as a hyperbolic system of conservation laws based on the total variation diminishing (TVD) scheme. Several standard tests and test simulations are presented to demonstrate the accuracy, robustness and flexibility of the code. Using this code we have studied three-dimensional hydrodynamic interactions of relativistic extragalactic jets with two-phase ambient media. The deflection angle of the jet is influenced more by the density contrast of the cloud than by the beam Mach number of the jet, and a relativistic jet with low relativistic beam Mach number can eventually be slightly bent after it crosses the dense cloud. Relativistic jet impacts on dense clouds do not necessarily destroy the clouds completely, and much of the cloud body can survive as a coherent blob due to the combination of the geometric influence of off-axis collisions and the lower rate of cloud fragmentation through the Kelvin-Helmholtz instability for relativistic flows. We find that relativistic jets interacting with clouds can produce synchrotron emission knots similar to structures observed in many VLBI-scale radio sources and the synchrotron emission peaks right before the jet passes through the cloud.