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Federated learning (FL) is a promising paradigm to enable privacy-preserving deep learning from distributed data. Most previous works are based on federated average (FedAvg), which, however, faces several critical issues, including a high communication overhead and the difficulty in dealing with heterogeneous model architectures. Federated Distillation (FD) is a recently proposed alternative to enable communication-efficient and robust FL, which achieves orders of magnitude reduction of the communication overhead compared with FedAvg and is flexible to handle heterogeneous models at the clients. However, so far there is no unified algorithmic framework or theoretical analysis for FD-based methods. In this paper, we first present a generic meta-algorithm for FD and investigate the influence of key parameters through empirical experiments. Then, we verify the empirical observations theoretically. Based on the empirical results and theory, we propose a communication-efficient FD algorithm with active data sampling to improve the model performance and reduce the communication overhead. Empirical simulations on benchmark datasets will demonstrate that our proposed algorithm effectively and significantly reduces the communication overhead while achieving a satisfactory performance.