Confused about resistors and potential dividers?

I thought that resistors worked by increasing the difficulty with which the electrons move through that part of the circuit. The electrons had to "fight" to get past the resistor, and in doing so used up a certain amount of electromotive force (which, I've been told, is equivalent to voltage). However I am studying potential divider circuits and it is said that the voltage, rather than being used up to "get across" the resistor, goes into a side-circuit and can be used used to operate, for instance, a bulb. If the resistor used is a variable resistor and the resistance decreases, it that there is less voltage travelling into the side-circuit so the bulb doesn't light. I don't understand this; if there is less voltage being used up to "get across" the resistor surely there is more voltage left to travel into the side-circuit? Please help!!