Ok, Ballscrews and ballnuts. The basic idea is that you need to convert rotary motion (a motor turning) into linear motion (a car rolling down a rail). There are a few ways to do this: You can use a belt, or put little wheels on a track, or use a threaded rod and nut. In the case of a hardened CNC machine like this you need it to be able to provide a large amount of force without getting nudged around or flexing, so that last option, the threaded rod, is a pretty good candidate.
A regular threaded rod will work sometimes, but in general they are very fragile because the edges of the threads are pointy and easy to get chips or bends in them. The simple solution to this is make threads that are flat on the outside so that they are less fragile. This is called an "Acme thread."
There's one step better though, which is to get a very special threading that uses ball bearings to roll on. The advantage of this is a lower friction (meaning more power makes it from rotary force to linear force). In some cases ballscrews can be twice as efficient as an Acme threaded rod. They are, however, generally between 4 and 20 times more expensive depending on just what you get. You do save money on motors, controllers, and power supplies, but it's not always worth it. Do the accounting for your application, and feel free to ask me questions. I'll be happy to elaborate if you're looking to build one of these.
Here's a neat animation of how a ballnut works; notice that the ball bearings have to be recirculated through the nut in order to keep rolling on the screw:
On to the design...