I'm just getting into 3-D artwork and more specifically, Blender. I am currently just trying to download the program, and on the download page it gives me the option to download from "Mirror" links. Do these separate links carry the same files and if not, which should I use?
A mirror is like a backup download link. If one for some reason goes down you can still download the file. They all contain the same file so it doesn't really matter which one you choose.
While all of them will give you the same file, some will be closer to you (geographically) than others.
All else being equal, you'll get a better download speed with a server that's nearby.
Try the first, if it's slow try the second, and so on, they are all the same file.