The turbulent structures formed in a Taylor-Couette (TC) flow established between two concentric counter-rotating cylinders were explored numerically. The shear Reynolds number was set to Re-shear = 8000 and the radius ratio was set to r(i)/r(o) = 0.5. An optimal flow corresponding to the maximal angular velocity transport between the cylinders was selected for the TC flow. The mean velocity profile reached its steepest value near the cylinders in the optimal TC flow. The streamwise velocity correlations at the outer cylinder in the gap exceeded those at the inner cylinder. The large convective transport of angular velocity in the gap generated a maximal angular velocity flux to achieve the optimal flow. The angular velocity flux generated by the momentum source exceeded that generated by the momentum sink. The vorticity dispersion was larger near the inner cylinder than near the outer cylinder, but vorticity stretching near the outer cylinder exceeded than that near the inner cylinder. The skin friction coefficient budgets were plotted using the velocity-vorticity correlation. The vortex stretching contributions dominated the skin friction budgets. The area near the inner cylinder was populated by stronger vortices, but their population density was smaller than the population density of the vortices near the outer cylinder. The probability density functions of the wall-normal and streamwise velocity fluctuations delineated the presence of the large wall-normal velocity fluctuations near the outer cylinder.