You probably want to read about biconnected components first; these are just a wrinkle on top of biconnected components.
A connected component can be viewed both as an equivalence class of edges of a graph and as an equivalence class of vertices of that graph.A biconnected component is an equivalence class of edges of a graph. But if we look at the vertices belonging to those edges we don't get an equivalence class; see the diagram at biconnected components for an example.
A vertex belonging to more than one biconnected component is an articulation point of the graph. If we remove an articulation point from the graph, it falls apart into several connected components.