Connectivity and components
Intuitively, a graph is connected if you can't break it into pieces which have no edges in common. More formally, we define connectivity to mean that there is a path joining any two vertices - where a path is a sequence of vertices joined by edges. The example of $Q_3$ in your question is obviously not connected - none of the vertices in the bit on the left are connected to vertices in the bit on the right. Alternatively, there is no path from the vertex marked 000 to the vertex marked 001.
(As an aside - all trees are connected - a tree is defined as a connected graph with no cycles. But there are many other connected graphs.)
So if a graph is not connected, then we know it can be broken up into pieces which have no edges in common. These pieces are known as components. The components are themselves connected - they are called the maximal connected subgraphs because it is impossible to add another vertex to them and still have a connected graph. All connected graphs have only one component - the graph itself.
You can think of the cut induced as being the set of edges which connect some collection of vertices to the rest of the graph. In the diagram you give, the set called $A$ is the collection of vertices within the dotted line. The cut induced by $A$ is then the collection of edges which cross the dotted line - the edges which connect the vertices inside the dotted line to those outside it. Edges joining vertices inside the shaded area are not part of the cut induced, and neither are edges joining vertices on the outside of the dotted line.
More formally, the complement of $A$ is exactly those vertices which are not in $A$. So the cut induced by $A$ is the collection of edges joining vertices in $A$ to vertices in the complement of $A$.