This is a very broad subject, involving the mathematical processing and/or analysis of sampled signals. Applications include:
Digital filtering – processing sampled signals to remove unwanted noise and interference. Signals can be 1D (such as audio) or 2D (an image) or higher (another time ;o)
Spectrum Analysis – transforming signals that are sampled in time to the “frequency domain” allows data to be viewed and analyzed in alternative ways. Astronomers, chemists, physicists, medics and engineers all use spectrometers to detect or visualize different phenomena.
Digital Control – Electronic control of a mechanical device, be it a heater in a building or the speed of a motor in a wind turbine, all need to take measurements of feedback parameters (such as temperature and rotational velocity) so that timely real-time adjustments can be made. Many modern systems use digital sampling to digitize feedback data, perform often complex analysis in software, before adjusting the outputs to control a device. The AirBus fly-by-wire systems are an example of safety-critical digital control systems.
Monitoring and instrumentation – Similar to control in that signals are measured, but where the information is digested by human (and sometimes make adjustments or decisions). Examples include heart monitoring, alarms, automotive instrumentation etc.