I think I'm getting confused as to how these things work. I thought they were like a computer run milling machine, but this seems to be something a bit different.
They're completely different. A 3D milling machine removes material, while 3D printers add.
There are 3D printers that can print directly in metal or other non-plastic materials, but let's focus on the printers within hobbyist reach:
Most of the cheap low end printers (like the Dremel, above) are extrusion printers (FDM - fused deposition modeling) and typically use a thermoplastic material (like ABS) heated in a nozzle. The printer deposits layers of molten plastic to build up the model. The results will usually have noticeable steps that will require effort to remove or conceal.
Selective Laser Sintering (SLS) fires a laser into an easily melted powder, like nylon. The small particles stick together, layer by layer, until the model is completed. The results have a grainy finish. You cannot sand the model to eliminate the graininess, because it's grainy all the way through. The particles are just stuck together. You can soak it in paint or primer and sand that.
Color printing can be done with a plaster-like powder and special inkjet heads (Powder bed Printing). The result is grainy (like plaster) and fragile, but you can get a nice color representation.
Stereolithography fires a laser (usually UV, but not always) into a vat of photosensitive resin. The model is built up layer by layer. Stereolithography has the advantage of extremely high resolution and many models will require little if any post processing.