I understand the Fourier transform pretty well, and that I'm finding the frequency domain of a signal in the time domain.
I also understand that the Fourier transform is a special case of the Laplace transform where $\sigma=0$ for $s=\sigma+j\omega$
If $\omega$ represents the frequency of the sinusoids that make up a time signal, then what does $\sigma$ represent?
Furthermore, why does that relationship change in the Z transform? The DTFT is a special case of the Z transform not when the real portion is zero, but when $r=1$ in $z=re^{j\omega}$, or when $|z|=1$. So in this case, what does the radius represent?
And what does one tell us over the other? If the Fourier transform contains all the information we need about a signal, why bother with Laplace or Z? If Z contains more information, why do we bother with Fourier?