When both types of filters are combined into one, that is, a capacitor and resistor in series is used as input and a capacitor and resistor are used in parallel for the feedback, a new type of filter emerges: the bandpass filter.
To see how this works, we need to simplify the circuit to use only one element instead of two, in order to make analysis easier. Since when AC is applied to a capacitor it can be replaced with its capacitive reactance in ohms, we can use that to combine it with the series resistor at the input, and with the parallel resistor for feedback.
This gives us an input impedance (Impedance is a generalization of resistance that also includes reactances, and is also measured in ohms) and a feedback impedance, in a configuration similar to the simple inverting amplifier.
Since both impedances are frequency dependent, the gain will be frequency dependent as well. At low frequencies, the input capacitor's reactance is very high and dominates the series combination with the resistor, so the input impedance becomes very large. At the same time, the feedback capacitor will also have a very high reactance, but this time the resistor dominates because the connection is made in parallel.
Since the gain is defined by the ratio -Rf/Rin, generalized to impedances as -Zf/Zin, where Z denominates impedances in most electronics literature. Since the feedback impedance is small, limited by the resistor, compared the input impedance which tends to infinity, the ratio will be very small and will attenuate the signal (Zf << Zin, so the ratio is less than 1). In this case, the extremely high input impedance drives the ratio towards zero.
At very high frequencies, the input impedance is dominated by the resistance, since the capacitor's reactance is very small. The opposite effect happens at the feedback, since now the capacitor dominates with its very low reactance, which makes the impedance very low.
Checking the gain ratio -Zf/Zin, we can see that now the input impedance is very low, limited by the input resistor, but the feedback impedance will be lower still, going towards zero, not being limited by anything since the capacitor is dominating the connection, so the ratio will again be very small, attenuating the signal. This time, the very small feedback impedance drives the ratio to zero.
At medium frequencies, where no single component dominates each connection, both input and feedback impedance will be very close to each other, since they will be a very similar value, assuming equal components. At the frequency where the series combination and the parallel combination have the same value, the gain will be 1, given by the ratio -Zf/Zin, where Zf = Zin; This is called the center frequency, and it is the only signal that will not be attenuated.
The overall effect is that this circuit will attenuate both high and low frequency signals applied to it, and only pass a small range (also called band) of frequencies where both input and feedback impedances have a very similar value, hence the name bandpass filter. This is useful when you need to block noise or extra signals created within a circuit.
