I'm not a guru, but I'll do my best to give an overly vague explanation and completely gloss over the important details.

I sort of assume that you understand the basics of electricity already. (If not, no explanation of motorcycle electrical systems is going to make sense to you.) In particular you already need to know how electricity flows, the difference between voltage and current, etc.
There are
four main electrical systems on your bike: the starting system, the charging system, the ignition system, and what I'll call the accessories. The starting system engages the starter motor, the charging system charges the battery, the ignition system makes the spark plugs spark, and the accessories are things like lights that may be required for the bike to be road-legal, but are not essential to the running of the engine. (I'll ignore the accessories here.)
First, the starting system. Pretty simple. When the ignition key is turned on, the "start" button on the handlebars completes a circuit that closes a big relay called the
solenoid. When the solenoid closes (with a loud click), it completes another circuit between the battery and the starter. Which runs the starter and turns the engine over, starting it. (Hopefully.)
Next, the charging system. The charging system, as you might guess, is responsible for keeping the bike's battery charged. On almost every bike ever made, it has two main parts: the alternator, and the regulator/rectifier (commonly called the R/R). The alternator generates electricity from the rotation of the engine. But it generates
alternating current (AC). The battery needs
direct current (DC). So the R/R sits between the alternator and the battery and simply
rectifies (converts) the AC to DC. It also
regulates the amount of current going to the battery, so that the battery doesn't get more voltage than it can handle, no matter the RPM of the engine.
The "alternator" is made up of two parts: the rotor and the stator. The rotor is really just a bunch of magnets glued to the flywheel. The stator is a series of wire coils that generate current when the rotor passes by.
Make sense?
I'll try to dig up some references for the ignition system, since I'm not sure I can explain that one on my own...