In federated learning (FL) systems, e.g., wireless networks, the communication cost between the clients and the central server can often be a bottleneck. To reduce the communication cost, the paradigm of communication compression has become a popular strategy in the literature. In this paper, we focus on biased gradient compression techniques in non-convex FL problems. In the classical setting of distributed learning, the method of error feedback (EF) is a common technique to remedy the downsides of biased gradient compression. In this work, we study a compressed FL scheme equipped with error feedback, named Fed-EF. We further propose two variants: Fed-EF-SGD and Fed-EF-AMS, depending on the choice of the global model optimizer. We provide a generic theoretical analysis, which shows that directly applying biased compression in FL leads to a non-vanishing bias in the convergence rate. The proposed Fed-EF is able to match the convergence rate of the full-precision FL counterparts under data heterogeneity with a linear speedup. Moreover, we develop a new analysis of the EF under partial client participation, which is an important scenario in FL. We prove that under partial participation, the convergence rate of Fed-EF exhibits an extra slow-down factor due to a so-called ``stale error compensation'' effect. A numerical study is conducted to justify the intuitive impact of stale error accumulation on the norm convergence of Fed-EF under partial participation. Finally, we also demonstrate that incorporating the two-way compression in Fed-EF does not change the convergence results. In summary, our work conducts a thorough analysis of the error feedback in federated non-convex optimization. Our analysis with partial client participation also provides insights on a theoretical limitation of the error feedback mechanism, and possible directions for improvements.