difference equation

An equation defining each output sample from past inputs and (if recursive) past outputs -- the discrete-time equivalent of a differential equation.

A linear constant-coefficient difference equation expresses each output sample $y[n]$ as a weighted sum of current and past inputs, and -- if the system is recursive -- past outputs too, via that same recursion. Its general solution splits into a homogeneous part (the system's own natural response) plus a particular part (the response driven by this specific input).