Network structure analysis plays an important role in characterizing complex systems. Different from previous network centrality measures, this article proposes the topological centrality measure reflecting the topological positions of nodes and edges as well as influence between nodes and edges in general network. Experiments on different networks show distinguished features of the topological centrality by comparing with the degree centrality, closeness centrality, betweenness centrality, information centrality, and PageRank. The topological centrality measure is then applied to discover communities and to construct the backbone network. Its characteristics and significance is further shown in e-Science applications.