Gamma-ray spectroscopy will always create a fairly large Compton continuum consisting of gamma rays that undergo scattering within the detector followed by their escape. 

Simply stated, Compton suppression is a continuum reduction technique as can be seen in the two spectra of Figure 1. Without Compton suppression, excessive scatter from high-energy gammas can result in obscuring peaks with low branching intensities. The high purity germanium (HPGe) spectrum shown has a continuum reduction factor of six. In this example a peak to Compton ratio of 600:1 can be achieved for Co-60.  A HPGe spectrometer has high resolution but does not have a high Z (32), which further contributes to escaping photons.

Fig. 1. Background Reduction (Before and After)

Compton suppression generally applies to HPGe spectrometers but can be used in conjunction with high resolution, low background scintillators like cerium bromide (Z_eff = 45.9).