The key ideas for series and parallel resistors follow from the behavior of the current and voltage through each configuration.

For series resistances, the CURRENT through each resistor of the series is the same.

Think of it like a water pipe with smaller pipes along the length, which act as resistances to the flow of the water. If there are no leaks or new sources, all the water through one smaller resistance pipe must eventually flow through the other. Also, the voltage drop across each resistor individually must add to the total voltage across the whole series.

Take, for instance, this series of identical resistors of a value R. If a voltage is applied across the entire length, a current is generated through each resistor, the value of which is: (total voltage) / (total resistance). Now, each resistor will have that SAME current flowing through it, so the voltage difference across several resistors will change depending on how many resistors are measured across.

ie: For one resistor, you get V = i R. Two resistors, you get V = i(2R), which is twice what it was for one R. And Three R's, I get V = i(3R). Voltage difference gets larger if the total resistance gets larger because current is the same.