Within the federated learning (FL) framework, the client collaboratively trains the model in coordination with a central server, while the training data can be kept locally on the client. Thus, the FL framework mitigates the privacy disclosure and costs related to conventional centralized machine learning. Nevertheless, current surveys indicate that FL still has problems in terms of communication efficiency and privacy risks. In this paper, to solve these problems, we develop an FL framework with communication-efficient and privacy-preserving (FLCP). To realize the FLCP, we design a novel compression algorithm with efficient communication, namely, adaptive weight compression FedAvg (AWC-FedAvg). On the basis of the non-independent and identically distributed (non-IID) and unbalanced data distribution in FL, a specific compression rate is provided for each client, and homomorphic encryption (HE) and differential privacy (DP) are integrated to provide demonstrable privacy protection and maintain the desirability of the model. Therefore, our proposed FLCP smoothly balances communication efficiency and privacy risks, and we prove its security against “honest-but-curious” servers and extreme collusion under the defined threat model. We evaluate the scheme by comparing it with state-of-the-art results on the MNIST and CIFAR-10 datasets. The results show that the FLCP performs better in terms of training efficiency and model accuracy than the baseline method.