Vertex connectivity is the minimum number of vertices you must remove to disconnect the graph (or make it trivial if already small). This measures node-level resilience. You can compute it using max flow by splitting each vertex v into vin and vout with an edge of capacity 1 between them.
Then find minimum s-t cuts for all pairs. The split vertices enforce that removing a vertex counts as 1 unit. Vertex connectivity measures how many node failures the network can handle before becoming disconnected.