To the main topic, the actual use of resistors is not as interesting/useful as the concept of resistance. resistance = work using element. this gives rise to concepts like "resistance, as seen by X". eg, input and output resistances of an amplifier, which may be different than any physical resistor in the circuit. for example, a motor spins. applying a MECHANICAL load will cause the motor to draw more current. the 240V AC line stays fixed at 240V, thus the motor APPEARS to have a lower resistance. This is 100% correct -- the mechanical load uses energy that came from the 240V line!
eg, the input resistance of an emitter follower is possibly 100 times higher then the resistor used in the circuit. To apply a voltage across this resistor, the majority of current comes from ANOTHER power source (battery), and thus you have less current for a given voltage = higher inputer resistance.
and the output resistance of this circuit is also lower then any series resistance the source might have for the same reason -- if output current needed to increase, it would pull MOST of the current from the power supply NOT the source!
From here, ideas of the incremental resistance arise. This brings up the idea of "negative resistance". This only means that when the input signal increases, some OTHER power source forces a current back to the source. This applies to some oscillators where an RLC network (damped oscillator) is aided by adding some negative resistance to make the circuit into a full blown oscillator.
From there, the other amplifer types arise (transresistance - current to voltage, transconductance - voltage to current, voltage gain - voltage to voltage, current gain - current to current)
once concpets of reactance -- elements that store and release energy -- are added, you can start to see all kinds of interesting things.