I see many groups in ToA wanting to do a "chest run". My question is, what exactly is a chest run? Do you actually "run" from chest to chest? Or is it just another way to say "farming"?

You literally run from chest to chest. Don't ever go on chest runs, chances of you getting a good drop out of it is as much as getting struck by lightning. I went on 12 runs, used 7 keys, 6 of them came out worthless shadow staves and 1 was a req 9 chaos axe. (I was so happy when it dropped out of the box! And it turned out to be purple)