Bose-Einstein condensates (BECs) provide a testbed for a wide array of coherent structures with complex dynamical properties. Of these structures, vortices and two-component BECs are at the forefront in understanding fundamental properties of BECs and have been under intense scrutiny in both experiments and theoretical studies. The behavior of these structures elucidates the mechanics of nonlinear processes that give rise to patterns in vortex lattices and patterns in binary BECs. This has lead to the integration of BECs into the new field of emergent phenomena that has unified many seemingly unrelated disciplines because at a fundamental level, the nonlinear processes provide a blueprint to give rise to coherence out of randomness. First, we study the interactions between two atomic species in a binary BEC to determine conditions for miscibility, oscillations between species, steady state solutions and their stability. Second, the two component system is extended to a quasi-2D systems for a pancake-shaped condensate. Third, the shape of the background atomic density as well as the background with a vortex is studied to determine the role of the phase and background on the precession of a vortex. Lastly, the dynamics of small clusters of same charge vortices in a trapped BEC is studied giving fixed point configurations that rotate at a constant speed
