After a system has been defined mathematically (in terms of Transfer Function, State Space, ... or any other mathematical representation), it is then possible to determine how the system behaves as a function of time for a given input and initial conditions.  The response to the initial conditions alone (with no input) is called the zero-input response; the response due to the inputs alone (with initial conditions at t=0- equal to zero) is called the zero-state response.  The most commonly determined transient responses are those due to a step, an impulse and a sinusoid.  The transient response is closely linked to the frequency response and the poles and zeros of the transfer function.

The map above shows (among other things):