Just as the high pass filter is a variation of the low pass filter, changing the reactive element from input to feedback, so is the band stop filter a variation of the band pass filter, but instead of changing components we are going to change the configuration of the components.
For this circuit, the input impedance consists of a resistor and capacitor in parallel (it was in series for the band pass), and the feedback impedance will be a capacitor and resistor in series (was parallel in bandpass). As you can see, only the connections change, the components stay the same.
At low frequencies, the input impedance is dominated by the resistor, since the reactance is much higher than the resistance (the connection is in parallel, the equivalent is always lower than the lowest value). At the same low frequencies, the feedback impedance is dominated by the capacitor's reactance, since it is also high compared to the resistor (the connection is in series, the equivalent is always higher than the highest value).
The gain of the opamp, connected in an inverting amplifier configuration, is given by -Zf/Zin. The input impedance Zin is very low, near by the input resistance, and the feedback impedance is very high, driven by the capacitors reactance; this makes the ratio very high, tending towards infinity by the increasing Zf at lower and lower frequencies (it is theoretically infinite at DC, or 0hz frequency).
One way to limit the gain, similar to what was done for the low pass filter is to use a resistor in series with either the whole feedback series connection or just across the capacitor. This makes the extremely high reactance of the capacitor not dominate at very low frequencies, instead the parallel connection is closer to the lower value, in this case the resistor. This is done to ensure that the opamp does not go into saturation, because if it does it clips the signal and distorts it.
At very high frequencies, the input impedance tends towards, since the capacitor acts as a very low value. With the feedback connection, the capacitor is also a very low value, but since there's the series resistor, the impedance will be limited to that value.
Looking at the gain equation (-Zf/Zin), you can see that the gain tends towards infinity, since the input impedance goes very low at high frequencies. To limit this, you can put resistor in series with the original parallel combination.
At intermediate frequencies, where the input impedance and feedback impedance are very close, the gain will be close to 1.
With all this, you can see that the notch filter is the opposite of the band pass filter: the band stop filter highly amplifies signals above and below the "notch" or middle frequencies, and doesn't amplify (instead of blocking) the intermediate frequencies. This is in contrast with the band pass that attenuated signals above and below, and also didn't amplify intermediate frequencies (gain of 1).
For all the filters discussed so far there are other far more efficient and that also block undesired signals and amplify the frequencies of interest.
