A system transforms an input signal into an output signal. We can write this abstractly as

$$y=T{x},$$

where $T$ denotes the system, $x$ is the input and $y$ is the output.

The same idea applies in continuous time,

$$y(t)=T{x(t)},$$

and discrete time,

$$y[n]=T{x[n]}.$$

A system may represent a physical device, a differential equation, a digital algorithm or any rule that maps one signal to another.

The system description and the input should be kept conceptually separate. The same system can respond differently to different inputs, while the same input can produce different outputs when passed through different systems.

Signals describe quantities that vary; systems describe transformations of those signals.