Two graphs are isomorphic if you can re-draw one of them so that it looks exactly like the other.
To re-draw a graph, it helps to imagine the edges as infinitely stretchable rubber bands. You can move the vertices around and stretch the edges any way you like -- as long as they don't become disconnected.
Sometimes it is very hard to tell whether two graphs are isomorphic or not. In fact, no one knows a simple method for taking two graphs and determining quickly whether or not they are isomorphic.
