The Sixth Generation (6G) networks are designed to provide ubiquitous, customized and intelligent services for users. Federated Learning (FL) as a privacy-preserving Artificial Intelligence (AI) paradigm is expected to be a key enabler for realizing ubiquitous distributed AI in 6G networks. Given this, this paper mainly focuses on FL in 6G, combining the advantages of both FL and 6G to provide a novel perspective and promising paradigm for data collection, processing, transmission and intelligent applications. However, despite the fact that raw data of different users is stored locally, achieving efficient FL in 6G still suffers from a series of tough challenges in terms of heterogeneity, communication bottleneck, privacy and security risks. This paper firstly presents an AI-enabled 6G network with Space-Air-Ground Integrated Network architecture and FL. Furthermore, we discuss the heterogeneous challenges and potential threats faced by FL such as heterogeneity about device resources, data distribution, data modality, together with communication bottleneck as well as inference attack and backdoor attack in detail. Moreover, countermeasures and tensor-empowered lightweight representations for personalized FL are proposed to conquer the aforementioned challenges, where tensor train decomposition is leveraged to capture the low-rank structure of AI models in FL for reducing communication overhead as well as computing power and storage costs during model training and inference. Finally, experimental results show that the proposed approaches have outstanding performance, and some open issues and potential research directions for FL in 6 G are also emphasized.