What does 'z' in Z-transform represent ? Is it frequency or something else? my question is about the Z- transform. My first question is what the title says. What does 'Z' in Z-transform represent ? Say in Fourier transform, 'w' (omega) represents frequency ? From Fourier transform, I can know what is the strength of the signal at a certain frequency component, right? Then what idea do I get from Z-transform ?

And, in Fourier transform, we can draw a graph with x-axis as the frequency and y-axis as the amplitude of a certain frequency component. Can we do the same for Z-transform, with x-axis as the different 'Z' values and y axis as the Z-transform for a particular Z ?  My textbook says only about equations, but I don't get the significance of 'Z' in  Z-transform. And about the graph thing, I haven't encountered any such graph in any textbook, only the Z-plane.
The Fourier transform doesn't exist for every signal. For example, there is not a Fourier Transform of the signal $ (2)^{n} u(n) $. So, in order to get an useful toll for this signals, you can multiply the signal for a sequence $ r^{-n} $ in order to make it decay fast enough to converge:

$$ X(\omega) = \sum_{n=-\infty}^{\infty}{[x(n)r^{-n}]e^{-j \omega n}} $$

$$ X(\omega) = \sum_{n=-\infty}^{\infty}{x(n)(re^{j \omega})^{-n}} $$

Then, we can make $ z = re^{j \omega} $. So, in this case, z is a complex value that can be understood as a complex frequency.

It is important to verify each values of $ r $ the sum above converges. These values are called the Region of Convergence (ROC) of the Z transform. The z transform is equal to the Fourier transform when $ r = 1 $, but the Fourier transform will only exist if $ r = 1 $ is inside the ROC.

As $ z $ is a complex variable, you can't plot it in a Cartesian graph as you said. You will need to plot an 3D graph where x-axis and y-axis contain the real and imaginary values of $ z $, and the z-axis the values of the z-transform. I don't know if there is a real utility for a graph like that in any application, and that's why it is not common to see this graph. Normally, the z-plane is enough.