The concept of a graph is very simple to grasp, yet these mathematical objects are extremely varied. The specialized vocabulary for talking about graphs is most useful for trying to describe the various graphs and their properties. A Java applet for playing around with graphs. You can create and color graphs, find shortest routes and more. It is more or less obvious that the board of Sam Loyd's 15 puzzle might be abstracted to a 4x4 graph every counter position corresponding to a node.
Edges indicate possible puzzle moves, i.e. moves of the empty square.
Less obvious is that the graph is bipartite. An interactive game in which players take turns removing fruit from the table according to the rules. The player who removes the last fruit from the table loses. Based on the mathematical puzzler known as the game of Nim. A list of links to materials on the Web.