This is an extremely fundamental idea here.

Although not *quite* the right way to put it, the “time domain” and “frequency domain” (as signal processing people tend to put it) are like different bases of a vector space.

A “system” is a function from signals to signals. A “signal” most commonly a function mapping a[1] real value (often time) to a[1] real value (eg, voltage, or what have you). In a DSP context, the function maps from integers instead of reals…

In many of the simple contexts, particularly in DSP, the “systems” are actually a linear transformation.

[1] Or k-tuples also