I will explain the concepts in signal and systems terminology (which you can also import easily into control systems and communications).
Most of the systems which are picked up for study are <a href="https://en.wikipedia.org/wiki/LTI_system_theory" target="_blank" rel="nofollow noopener noreferrer">Lti System Theory</a>, i.e Linear Time Invariant. Various signals can be applied to these systems. A special class of signals if applied as input to such a system will have it's shape retained at the output, i.e their amplitude may change and their phase may change.But other than that there is no difference between input and output signals. (Such an input signal is an <a href="https://en.wikipedia.org/wiki/Eigenfunction" target="_blank" rel="nofollow noopener noreferrer">Eigenfunction</a> to the system and the amount by which it's amplitude changes is the <a href="https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors" target="_blank" rel="nofollow noopener noreferrer">Eigenvalues And Eigenvectors</a> of the system for that eigen function). For LTI systems a complex exponential signal is a typical signal which retains it's shape at the output i.e only the amplitude and phase may change. The 'transfer function' basically gives you an idea how the amplitude and phase may change for a particular exponential.
Suppose the 'transfer function' is H(s) and the complex exponential at the input is e^(s1t) (here s = s1) . The output is
|H(s1)| * e^(s1t - arg(H(s1))).
H(s) is complex function of s, so |H(s)| is its magnitude and arg(H(s)) is argument.
You can say that the 'gain' of the system is |H(s)|. As you will observe for different values of s (i.e for different complex exponentials) we get different gains.
Arg(H(s)) will be the 'phase change'.
We can't speak of gain as such for more general input signals as their shape is not conserved at the output of the system as is the case for exponential signals.
I Hope that some of your queries are cleared.
