How is electricity from the power supply turned into stored bits of data?
You might want to narrow down the question more instead of asking for all of Wikipedia.

But in general, "1s and 0s" depend on your system. Voltages are run through a series of comparators (usually grouped into some sort of ADC) which compare the input voltage to a reference voltage. If the input voltage is high enough, it's registered as a 1, if it's too low, then it's registered as a 0. You can read more about this at http://en.wikipedia.org/wiki/Logic_level

For example, assume on a given micro-controller or processor, the logic level high is 2.5V. So providing a pin with 3.3V would register that pin as being "high" or "1". 

These 1s and 0s are then stored in some sort of memory location and manipulated. The memory locations are implemented as "Latches". The physical implementation might vary slightly in size/complexity, but that's the general implementation.

Manipulations on these data bits can then be chained to create more and more complex operations. For example, a multiply operation is actually several addition operations.

This is really not a full blown textbook answer. You need ... a textbook for that. But this should get you started.