A series R-L-C circuit acts as a resonant circuit.A resonant circuit is one where only a particular frequency component is allowed and all other frequencies are cut-off.This type of circuits are typical to a band pass filter but the only difference being, the band pass region of a normal band pass filter is much higher compared to a resonant circuit.The range of frequency that a resonant circuit allows is theoretically only one frequency ((i.e) resonant frequency).
The design of these kind of circuits can be done if ω2 = [(Inductance Value)*(Capacitance Value)]-1.
Working:
The net impedance of a series R-L-C circuit is given as follows, Z=R+sL+(sC)-1. Any electrical component can be associated with a filter if it allows (parameter that drives a circuit)current to flow selectively through it.Now any filter is called a band pass filter if it allows current to flow for a specific range of frequencies.From the equation developed ,at resonant frequency the current flowing through the circuit is maximum as the the impedance provided by the circuit is minimum, at other frequencies the impedance is high.For further information about the analysis ,refer
A circuit that exhibits an effect called "resonance" can also be realized using an amplifier with 2nd order RC feedback.
However, independent on the realization (passive or active) you should know that such a bandpass circuit always has a certain bandwidth (frequencies near the resonant frequency are not "blocked", only attenuated)