Plane Settings of the Matched $z$-transform Method
The $s$-plane is the complex plane associated with the Laplace transform, i.e., with transfer functions of continuous-time systems, whereas the $z$-plane is the complex plane associated with the $\mathcal{Z}$-transform, i.e., with transfer functions of discrete-time systems. The matched $Z$-transform is one way of transforming continuous-time systems to discrete-time systems.

Take a look at this answer for more information about transformations from continuous-time to discrete-time.