A formal context is the data base and starting point for research in formal concept analysis. In order to describe the binary relations satisfying reflexivity and symmetry in a formal context, this paper firstly defines the symmetric formal context, and studies the characteristics of the concept lattice, the object concept, the attribute concept, the join irreducible concept and the meet irreducible concept of the symmetric formal context. Then, we propose the symmetric concept and prove that the symmetric concept set is a concept reduct of symmetric formal context. Finally, from the perspective of conflict analysis and social network, we show the practical semantics and application of the symmetric formal context, the symmetric concept and the symmetric concept set, respectively.