This work demonstrates the controlled motion and stopping of individual ferrofluid droplets due to a surface tension gradient and a uniform magnetic field. The surface tension gradients are created by patterning hydrophilic aluminum regions, shaped as wedges, on a hydrophobic copper surface. This pattern facilitates the spontaneous motion of water-based ferrofluid droplets down the length of the wedge toward the more hydrophilic aluminum end due to a net capillarity force created by the underlying surface wettability gradient. We observed that applying a magnetic field parallel to the surface tension gradient direction has little or no effect on the droplet's motion, while a moderate perpendicular magnetic field can stop the motion altogether effectively "pinning" the droplet. In the absence of the surface tension gradient, droplets elongate in the presence of a parallel field but do not travel. This control of the motion of individual droplets might lend itself to some biomedical and lab-on-a-chip applications. The directional dependence of the magnetoviscosity observed in this work is believed to be the consequence of the formation of nanoparticle chains in the fluid due to the existence of a minority of relatively larger magnetic particles.