The real-time dynamic environment perception has become vital for autonomous robots in crowded spaces. Although the popular voxel-based mapping methods can efficiently represent 3D obstacles with arbitrarily complex shapes, they can hardly distinguish between static and dynamic obstacles, leading to the limited performance of obstacle avoidance. While plenty of sophisticated learning-based dynamic obstacle detection algorithms exist in autonomous driving, the quadcopter's limited computation resources cannot achieve real-time performance using those approaches. To address these issues, we propose a real-time dynamic obstacle tracking and mapping system for quadcopter obstacle avoidance using an RGB-D camera. The proposed system first utilizes a depth image with an occupancy voxel map to generate potential dynamic obstacle regions as proposals. With the obstacle region proposals, the Kalman filter and our continuity filter are applied to track each dynamic obstacle. Finally, the environment-aware trajectory prediction method is proposed based on the Markov chain using the states of tracked dynamic obstacles. We implemented the proposed system with our custom quadcopter and navigation planner. The simulation and physical experiments show that our methods can successfully track and represent obstacles in dynamic environments in real-time and safely avoid obstacles. Our software is available on GitHub as an open-source ROS package.