When analyzing continuous-time linear systems, the Laplace transform is used to work with signals in the frequency domain. The Laplace transform can be thought of as the Fourier transform generalized to complex frequencies.
The transfer function H(s) of an LTI system represents how the system transforms the input signal exp(st). But what does such an input signal look like, and what is the meaning of a complex frequency?
Wonder no further. Play around with this complex frequency visualizer (continuous).
But what about discrete-time signals? We can sample signals at a period T and use the Z-transform to examine them in the frequency domain.
So what's the relationship between the s-domain and the z-domain? Check out this complex frequency visualizer (discrete).