Using a focused laser beam we stir a Bose-Einstein condensate confined in a magnetic trap. When the stirring frequency lies near the transverse quadrupolar mode resonance we observe the nucleation of vortices. When several vortices are nucleated, they arrange themselves in regular Abrikosov arrays, and in the limit of large quantum number the lattice structure is shown to produce a quantum velocity field approaching that for classical, rigid body rotation. Using a percussive excitation of the condensate, we measure the angular momentum of the condensate with vortices present and study the nucleation band as a function of the stirring intensity and geometry. We find that with only quadratic terms in the rotating perturbation the nucleation band is located around the quadrupolar resonance and has a width that increases with the strength of the stirring perturbation. However, when the potential includes cubic terms, the nucleation band broadens to include the hexapolar resonance as well. The results presented here demonstrate that the nucleation of vortices in the case of a harmonically trapped BEC is driven by the resonant excitation of the rotating quadrupolar mode, or by higher order rotating surface modes when the rotating perturbation includes the corresponding terms.