The longstanding goals of federated learning (FL) require rigorous privacy guarantees and low communication overhead while holding a relatively high model accuracy. However, simultaneously achieving all the goals is extremely challenging. In this paper, we propose a novel framework called hierarchical federated learning (H-FL) to tackle this challenge. Considering the degradation of the model performance due to the statistic heterogeneity of the training data, we devise a runtime distribution reconstruction strategy, which reallocates the clients appropriately and utilizes mediators to rearrange the local training of the clients. In addition, we design a compression-correction mechanism incorporated into H-FL to reduce the communication overhead while not sacrificing the model performance. To further provide privacy guarantees, we introduce differential privacy while performing local training, which injects moderate amount of noise into only part of the complete model. Experimental results show that our H-FL framework achieves the state-of-art performance on different datasets for the real-world image recognition tasks.