A notch filter decreases the amplitude of frequencies in the mid-range of the spectrum. It allows the low frequencies and the high frequencies to pass through. Audio engineers use this type of filter to cut out a problematic frequency.
A notch filter can be implemented by modifying the low-pass filter effect. Instead of using one sample of delay on the parallel path, the notch filter is created by using two samples of delay. A block diagram of the effect is shown below.
For high frequencies, like the Nyquist frequency, the two paths have constructive interference. The Nyquist frequency completes a cycle every two samples. Therefore, a two sample delay shifts the signal by a complete cycle, so it lines up with dry path.
For low frequencies, the two paths also have constructive interference. Since low frequencies have many samples per cycle, two samples of delay produces a negligible phase shift. When the delayed path is added to the dry path, they are essentially lined up with each other.
There is another interesting frequency to consider for this effect. It is the frequency which completes a cycle every four samples. Therefore, a two-sample delay shifts that frequency by half of a cycle. When the delayed signal is combined with the dry path, this frequency will have a phase difference of 180 degrees between the paths. This creates destructive interference for this frequency to cancel it out.
If the period of a cycle is four samples, then it is twice as long as the Nyquist frequency. In other words, the frequency is half of the Nyquist frequency, located in the middle of the spectrum. This is the frequency of the notch for the filter.
A complementary spectral effect to the notch filter is the band-pass filter. It reduces the amplitude of low frequencies and high frequencies, while letting a band of frequencies in the middle of the spectrum to pass through.