Think of a machine as a hierarchy. For example, when the slide on a lathe moves back and forth, the turret moves with it in lock-step. The turret is considered to be a child and the slide is considered to be a parent. In this manner, if the G-code specifies a certain movement, you can assign that movement to the parent and all of the children move with it. That way you don't need to specify how every piece of the machine moves independently.
You assign a piece of the machine to a particular movement. For example, you could assign a slide on a lathe to Z movement. So whenever the G-code has a Z move, the slide moves accordingly. If the slide is a parent of a turret, the turret moves too.
A milling machine has a top-most table that the fixture, vise, or stock from the FM file must mate to. A turning machine has a similar concept with the main-spindle chuck.
FeatureCAM users are familiar with the term UCS (User Coordinate System). UCSs are critically important in machine design. The term LCS (Local Coordinate System) is sometimes used and is equivalent to UCS.
A mate is a term from assembly modeling (or constraint-based positioning). Suppose there are 2 similarly sized cubes in space, randomly oriented and you want cube1 to be totally coincident with cube2 so that a specific corner of each cube occupies the same point as a specific corner of the other cube, and the edges are parallel and, in fact, coincident. Suppose further that there is a UCS that describes the position of the interesting corner of each cube and the orientation of the edges of that cube. We want to mate the UCS of cube2 to that of cube1, that is, we find a transformation (translation and rotation) such that the origins of the UCSs are made to be concentric, and the X, Y, and Z directions are the same. That is, there is one transformation (and only one) which takes cube1 (or cube2) and moves it to where you want it.