This article proposes a Novel Nonlinear Model Predictive Control (NMPC) for navigation and obstacle avoidance of an Unmanned Aerial Vehicle (UAV). The proposed NMPC formulation allows for a fully parametric obstacle trajectory, while in this article we apply a classification scheme to differentiate between different kinds of trajectories to predict future obstacle positions. The trajectory calculation is done from an initial condition, and fed to the NMPC as an additional input. The solver used is the nonlinear, non-convex solver Proximal Averaged Newton for Optimal Control (PANOC) and its associated software OpEn (Optimization Engine), in which we apply a penalty method to properly consider the obstacles and other constraints during navigation. The proposed NMPC scheme allows for real-time solutions using a sampling time of 50 ms and a two second prediction of both the obstacle trajectory and the NMPC problem, which implies that the scheme can be considered as a local path-planner. This paper will present the NMPC cost function and constraint formulation, as well as the methodology of dealing with the dynamic obstacles. We include multiple laboratory experiments to demonstrate the efficacy of the proposed control architecture, and to show that the proposed method delivers fast and computationally stable solutions to the dynamic obstacle avoidance scenarios.