Social Network Analysis (SNA) primarily studies the networks with graph-theoretic tools modeling individual actors within the network as the nodes and their relationships as the links of a graph. One of the SNA fundamentals is the notion of point centrality which was designed to reflect the relevant social significance among actors and has lately come to the attention of computer scientists; the latter seeks to identify highly central network nodes that can be exploited in the design of efficient protocols. This work studies the concept of centrality along three lines: Our first contribution is then an exhaustive survey and categorization of centrality indices that have been proposed over the last 40 years or so, along several attributes including the type of information (local vs. global) and processing complexity required for their computation. We next study the seven most popular of those indices in the context of Internet vulnerability to address issues that remain underexplored in literature so far. First, we carry out a correlation study to assess the consistency of the node rankings those indices generate over ISP router-level topologies. For each pair of indices, we computed the full ranking correlation, which is the standard choice in literature, and the percentage overlap between the k top nodes. Then, we let these rankings guide the removal of highly central nodes and assess the impact on both the connectivity properties and traffic-carrying capacity of the network. Our results confirm that the top-k overlap predicts the comparative impact of indices on the network vulnerability better than the full-ranking correlation. Importantly, the locally computed degree centrality index approximates closely the global indices with the most dramatic impact on the traffic-carrying capacity; whereas, is approximate power in terms of connectivity is more topology-dependent.