The goal of community detection algorithms is to identify densely-connected units within large networks. An implicit assumption is that all the constituent nodes belong equally to their associated community. However, some nodes are more important in the community than others. To date, efforts have been primarily driven to identify communities as a whole, rather than understanding to what extent an individual node belongs to its community. Therefore, most metrics for evaluating communities, for example modularity, are global. These metrics produce a score for each community, not for each individual node. In this paper, we argue that the belongingness of nodes in a community is not uniform. The central idea of permanence is based on the observation that the strength of membership of a vertex to a community depends upon two factors: (i) the the extent of connections of the vertex within its community versus outside its community, and (ii) how tightly the vertex is connected internally. We discuss how permanence can help us understand and utilize the structure and evolution of communities by demonstrating that it can be used to -- (i) measure the persistence of a vertex in a community, (ii) design strategies to strengthen the community structure, (iii) explore the core-periphery structure within a community, and (iv) select suitable initiators for message spreading. We demonstrate that the process of maximizing permanence produces meaningful communities that concur with the ground-truth community structure of the networks more accurately than eight other popular community detection algorithms. Finally, we show that the communities obtained by this method are (i) less affected by the changes in vertex-ordering, and (ii) more resilient to resolution limit, degeneracy of solutions and asymptotic growth of values.