This guide is not going to go into why one should or should not configure the pickups in their guitar in a series configuration. There are many qualified people who have written about this and have shared it on the internet. What this guide is going to do is take a look at it from an electronics point of view and briefly give a high-level explanation of what is happening in the circuit. My hope is that this will help to remove confusion regarding how it works. This is not written for the electrical/electronic technician or the engineer, but rather enthusiasts of whom are simply trying to wrap their minds around the concept.
Please note that the portion of the circuit that includes the connector and knobs and accompanying circuitry is not included in the diagrams below. This guide is only in reference to the part of the circuit that includes the pickups themselves.
Also note that this circuit utilizes a Single Pole Double Throw (SPDT) (ON, OFF, ON) Switch to change between three different modes of operation. Here is an explanation how each one works.
Look at diagram A below. Here the toggle is swept to the left position forcing the switch to make a connection between pins two and three in the diagram. Pin 1 is now disconnected. The electrical connection now creates a path that travels from the volume knob, through the switch and hits pin 1. Here it continues down to the left (back) pickup and then back up to pin 2 of the switch, through pin 3, and then on to be grounded.
You may have noticed that there are still wires coming from Pin 2 down to the second pickup and back up to pin 3. This is where you may asking yourself why the second pickup is not working as it is connected to the circuit. This is where the confusion usually stems from.
Here is one way to look at this. Electricity always wants to get to the ground utilizing the path of least resistance. Even though that pickup is connected to pins 2 and 3, the switch creates a point of contact between them. This creates a fork in the road. The pickup is seen by the circuit as resistance and therefore the electricity would rather travel through the switch instead. This "Turns off" that pickup.
Another way to think about it is from the pickup point of view. It does its job by creating a difference in potential in the wires connected to it. When the switch is closed between 2 and 3, this makes them electrically the same point. Since they are electrically the same point, any potential created is quickly balanced out between the two sides of the pickup and thusly eliminating it from the circuit.
Circuit C works the same as Circut A. It simply cuts off the other pickup.
In Circuit B however, the switch is in it's "OFF" position. This means that there are not any connections within the switch that would give the electricity a shortcut to ground. It is forced to go it to through both pickups, the first running through the second.