Online social media has developed rapidly in recent years, and many massive social networks have emerged. Traditional community detection methods are difficult to deal with these massive networks effectively for requiring knowledge of the entire network. Local community detection can find out the community of a given node through the connection relationship between the nodes around the given node without knowledge of the entire network structure, so it is of great significance in social media mining. For the relations between pairs of nodes in real-world networks are fuzzy or uncertain, the similarity relationship between two nodes with fuzzy relation is firstly described, and similarity between nodes as membership function of the fuzzy relation is defined. Then, it is proved that the fuzzy relation is a fuzzy similarity relation, and local community is defined as the equivalence class of the given node about fuzzy similarity relation. Moreover, local community of the given node is discovered by adopting maximal connected subgraph approach. The proposed algorithm is evaluated on both synthetic and real-world networks. The experimental results demonstrate that the proposed algorithm is highly effective at finding local community of the given node, and achieves higher F-score than other related algorithms.