When describing such a complex thing as a micro-processor it is hard to decide where to start. A good starting point could be the humble transistor.
The origin of the transistor can be dated back to 1926 but the transistor as we know it today was first produced at Bell Labs in 1959.
At its core, the transistor is an electronically controlled switch. If a conductive material like a copper wire is connected on one end to a current source, like a battery, and the other end is connected to ground, the wire will carry current. Put a light bulb in between and it should light up. If a mechanical switch is also introduced onto the wire, the current can be cut off from the light bulb by severing the physical connection between the current source and the light bulb, turning the light bulb on and off "by a flick of a switch".
The transistor works in a similar way, but rather than having to turn the switch on and off by hand, the transistor can be turned on of off by an additional source of current.
Transistors come in two flavors: NPM and PNP. Drastically simplifying things, the NPN transistor can be described as follow:
The transistors has three legs or terminals. The collector is connected to the current source. the emitter is connected to ground and the base is connected to the source that will control if the transistor is on or off.
Inside the transistor is an obstruction or a valve held in place with a spring. The spring will make sure that no current flows from the collector to the emitter. This is the off state. When current is introduced to the base, it will push the valve away, allowing current to flow from collector to emitter. This is the on state. This is the NPN transistor. By default the NPN transistor is in the off state.
This interactive illustration demonstrates the "mechanics" of the NPN transistor. In the beginning the electrons are trapped in the upper half of the transistor, in the collector half. Push the button and keep is pressed for a few seconds. Current from the base will start to flow causing the valve to open which will allow current to flow from the collector to the emitter. Release the push-button and the valve will close, cutting off current to the emitter.
The NPM's twin brother is the PNP transistor. It works in the opposite way. While current from the base opened up the NPM transistor, current from the base in a PNP transistor keeps it closed.
Current is now flowing both from the collector and the base, keeping the valve stationery. If current is now removed from the base, the force from the collector will push the valve to the left which allows current to flow from collector to emitter.
This is demonstrated in the illustration below. Both collector and base have electrons flowing indicating that there is a force holding the valve in place: top to bottom and from left to right. Press and hold the button. This will cut of current from the base. The current in the collector is so strong that it will push the valve to the left allowing current to flow to the emitter.
In reality, there are now springs or other mechanical components inside the transistor. Three layers of semi-conductive materials are sandwiched together to form the collector, emitter and base. The physical properties of these materials allow for the transistor to behave in the way described previously. The semi-conductive materials are either negatively charged (which is the N) or positively charged (which is the P).
The illustration below shows the three materials in the NPM or the PNP configuration. Next to them are the symbols used in electronic schematics.
This is what makes the transistor such a game-changer. Because there are no mechanical components inside the transistor, the transistor will never wear out. No springs will snap and no cog-wheels need to be replaced. The transistor can be made extremely small. So small in fact that many millions of transistors can be placed on a surface smaller than a fingernail.