It would help if you posted the spec of the machine in question.
Very simplified CNC System……………..
At its heart Mach3, 4, is an interface, trajectory planner, and G-code interpreter, however it can also be a motion controller in its own right, if you utilise the printer port(s), but this has it limitations.
A controller can include Mach3 as the interface but is actually a separate piece(s) of electronics kit that controls your Stepper and or Servo motors by receiving step and direction commands from Mach, it can also control relays, spindle etc etc etc.
CAD is where you create the drawings or 3D models.
CAM Software converts your 2D drawings or 3D cad model data into cutter path data.
Once the cutter paths are created they need to be output in a language your controller can understand, in Mach3's case G-code, this is done via a post processor, Mach3 closely emulates Fanuc, G-code, in my experience most Cam software will come with a standard Fanuc post processor, however its better to use a dedicated post processor for Mach3, some of which can be downloaded from the Mach download site, some Cam packages now come with a dedicated one for Mach, Fusion 360 being one of them.
HTH
Shaun