A CNC router machine is probably the most useful tool a hobbyist can own, but the price for a CNC machine on the market is way more than the average hobbyist is willing to spend. You can build your own CNC with very basic tools, little knowledge of machinery, mechanics, or electronics, but be warned, these machines are inherently dangerous, so wear the proper protection and use common sense. At the very least, read the instructions and precautions on every tool you use.
Now, a CNC stands for Computer Numerical Control, but that is gibberish. It does hint you to the fact that something is controlled, and probably by a computer. It's really a mechanism that interfaces to a computer to control its movements, similar to a robot, but this machine has a specific function. It cuts or shapes things with this control. In essence, a computer precisely controls a cutting tool, like a router for instance, to cut materials such as wood, plastic or metals into shapes only bounded by your imagination.
These two video tutorials will show you how to build the z-axis for a CNC router. What is a z-axis you ask? Do you remember 4th grade geometry? You only remember using the x and the y axis in that class, but it's the one in the 3rd dimension. Still confused? Ok... for the sake of simplicity, use a CNC machine as an example. Pretend you are looking at a router... got the image in your head? Ok, imaging that the router is moving up and down... up towards the sky and down towards the ground... all with the bit pointing downwards of course. And if you didn't know, the bit is the thing in the router that spins and cuts stuff. Well, the up and down motion is motion on the z-axis. The x axis moved forward and backward, and the y axis moves left and right. That's it.