Vertex Connectivity: The minimum number of nodes of a graph G, whose deletion from the graph G disconnects it. Vertex connectivity is sometimes called "point connectivity" or simply "connectivity."
In the above graph, if we delete $ \{ v4 \} $ or $ \{ v1, v5, v6 \} $ then graph will be disconnected. But we have to choose minimum number of vertex, hence we choose $ \{ v4 \} $.
Edge Connectivity: The minimum number of edges of a graph G, whose deletion from a graph G disconnects G, also called the line connectivity.
In the above graph, if we delete $ \{ k \} $, or $ \{ a, c, h \} $ or $ \{ a, c, d, f \} $ then graph will be disconnected. But we have to choose minimum number of edges, hence we choose $ \{ k \} $.
Separable Graph: A graph G is said to be separable if it is either disconnected or can be disconnected by removing one vertex, called articulation. A graph that is not separable is said to be biconnected (or nonseparable).
Yes, The above graph is a separable graph, because we can make it disconnected by deleting $ \{ v4 \} $. Hence $ \{ v4 \} $ is a cut-vertex or Articulation point.