The Z-plane is used by engineers to calculate quantities that are associated with alternating current, such as impedance. Electricity flowing through AC circuits has characteristics that are best represented by the two parts of a complex number, the kind of number plotted by the Z-plane.
The Z-plane is also known as the complex plane, because it is made up of complex numbers. Complex numbers are figures that consist of two parts: a "real" number, such as 6 or pi, and an "imaginary" number, such as the square root of negative 1.
AC current frequently changes directions, and engineers need to be able to model that accurately. Electrical engineers use the Z-plane for calculating properties of AC circuits because the two parts of a complex number allow for a simple mathematical way to represent the dual directions and amplitudes of alternating current.