Bose-Einstein condensates (BECs) are a quantum state of dilute atomic gases of weakly interacting bosons confined in an external potential and cooled to temperatures near absolute zero. We consider Gross-Pitaeviskii Equation (GPE), which is a variant of the Nonlinear Schoroedinger Equation (NLSE), to describe the dynamics of vortices in one-component BECs. By expressing a solution of the wave function in polar coordinate, an ordinary differential equation (ODE) is obtained, which is a well known vortex amplitude profile. This ODE is solved numerically and we use this numerically solved vortex profile to sit vortices on BECs. In order to solve GPE, we use 2nd order central finite difference scheme for spatial derivation and 4th order Runge-Kutta method to integrate in time. With different external potential, we observe different dynamics of vortices in BECs. With an infinite plane potential with a certain angle, the gradient of the field initiates vortex dynamics and a vortex moves in perpendicular to the gradient of the field. When the plane potential has no angle and have two vortices sitting on BECs, the change of phase and density difference in background initiate vortex dynamics. If they have same charge, they move around in circular motion; whereas, if they have opposite charge, they move in parallel to each other. With a magnetic trap, we use Thomas-Fermi (TF) limit (ie. Large atom number limit) to approximate the background density of the BEC. In this case, the gradient of the field initiates the precession of a vortex orbiting about the center of TF cloud. Our computational analysis observes all different vortex dynamics, and it confirms and supports theoretical analysis that has been done previously.