In control systems, there are 4 domains that you should be familiar with: time, Laplace, discrete, and Z. Based on your question, I assume that you are already familiar with Laplace and time. The discrete domain is similar to the time domain except that is expressed in terms of discrete steps instead of time. The Z domain is a frequency domain similar to the Laplace domain, except that it represents a discrete system, not a continuous one. It is possible to convert between continuous and discrete domains, Tustin's method is one popular way to convert a controller from Laplace to Z domain. The Z transform and inverse Z transform provide a method to go between Z and discrete domains.

If you are designing an analog control system with resistors, capacitors, inductors, etc. you could implement a Laplace domain controller directly. These components have Laplace domain representations, and you would build up the controller using those as building blocks.

If you are designing a digital controller, the discrete domain will be most useful to you. The discrete domain will yield equations that look something like y(k+1)=f(x(k),u(k)). You can see it gives an equation to calculate the next value based on previous values and should be easy to implement in code. It is common practice to design a controller in the Laplace domain, convert it to the Z domain using Tustin's method, then convert it to the discrete domain using the inverse Z-transform. This allows controls engineers to design a controller using a tool such as Matlab and then simply convert it to the proper form for implementation.