most likely refers to a filter design method for linear phase filters. half-band filters are very popular because they are very efficient. for a linear phase lowpass filter, the coefficients fall along a sinc function (or a sinc function that has been modified) The key property is that nearly every other sampled value is 0 for the half-band filter. eg, the coefficients might be 1 0 -2 0 4 0 -8 0 16 64 16 0 -8 0 4 0 -2 0 1. (I know these values have some issues, it's an example). this allows for a reduction of multipliers by around a factor of 2. This type of filter also becomes very efficient for decimate/interpolate by 2 applications.
Mth band filter expand on this concept by getting a 0 coefficient in every M'th location. This is done for cases where the bandwidth should be less than 1/2.