When I learnt the Laplace Transform I was just told the very standard formula that: $F(s) = \int_{-\infty}^{\infty}f(t) e^{-st} dt$. From this we went on to the table of transforms at its properties and how it is used. Later on we were told that $s=\sigma+j\omega$ and after that when we learnt Z-transform and we were told that the two are related via Discrete Time Fourier Transform.
We were taught that in control theory Laplace Transform helps us to find the poles and zeroes of a system transfer function and find out a lot about its dynamic behaviour. We were also taught how the Z-transform is used in expressing $x[n]$.
Now the problem; We have never been taught what $s$ really means and how it is able to do all these things, why $s=\sigma+j\omega$ and what it really means when we say that it is related to the Z-transform.
Now my question as already stated, what is this magic $s$ i.e what are its origins?