This paper presents a nonlinear model predictive controller to follow desired 3D trajectories with the end effector of an unmanned aerial manipulator (i.e., a multirotor with a serial arm attached). To the knowledge of the authors, this is the first time that such controller runs online and on board a limited computational unit to drive a kinematically augmented aerial vehicle. Besides the trajectory following target, we explore the possibility of accomplishing other tasks during flight by taking advantage of the system redundancy. We define several tasks designed for aerial manipulators and show in simulation case studies how they can be achieved by either a weighting strategy, within a main optimization process, or a hierarchical approach consisting on nested optimizations. Moreover, experiments are presented to demonstrate the performance of such controller in a real robot.