DSP stands for Digital Signal Processing and is a way to create filters and other signal processing functions in software rather than analog hardware. For a hearing aid it would be combined with an ADC and a DAC so the sound s are picked up by a microphone, turned into digital by the ADC, processed by the DSP to boost the amplitude of sounds you have less acuity to, conver ted back to analog by the DAC then played through the earpiece to make soun ds in your ear canal.
The big advantage is how these days a tiny digital device can perform a lot of processing and so implement very complex filters in a *very* tiny size and low power compared to analog techniques. In fact, DSP can do a number of things that are nearly impossible with analog.
Rick C.