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Distributed deep learning (DL) has become prevalent in recent years to reduce training time by leveraging multiple computing devices (e.g., GPUs/TPUs) due to larger models and datasets. However, system scalability is limited by communication becoming the performance bottleneck. Addressing this communication issue has become a prominent research topic. In this paper, we provide a comprehensive survey of the communication-efficient distributed training algorithms, focusing on both system-level and algorithmic-level optimizations. We first propose a taxonomy of data-parallel distributed training algorithms that incorporates four primary dimensions: communication synchronization, system architectures, compression techniques, and parallelism of communication and computing tasks. We then investigate state-of-the-art studies that address problems in these four dimensions. We also compare the convergence rates of different algorithms to understand their convergence speed. Additionally, we conduct extensive experiments to empirically compare the convergence performance of various mainstream distributed training algorithms. Based on our system-level communication cost analysis, theoretical and experimental convergence speed comparison, we provide readers with an understanding of which algorithms are more efficient under specific distributed environments. Our research also extrapolates potential directions for further optimizations.