And the word is antimacassar.
It's what you call that apron of extra fabric on the arm or back of a chair or sofa. To learn more about the history of these things check out this blog post.
When I bought this vintage chair for my one room challenge, the antimacassars on the arms were dirty and threadbare. Which is exactly why they serve such a good purpose: without them, the fabric on the arms would have been damaged, requiring re-upholstery. Instead, I was able to fix up the chair by simply removing the antimacassars.
In the store, I had figured that these were affixed to the chair with thread, as most antimacassars seem to be. However, once I got the chair home, I realized the antimacassars were held on with a metal spring! I didn't need wire cutters, though. I was able to sort of unscrew the spring from the chair and gently pull it out without damaging the fabric.
Antimacassars. Something to keep in mind when you find a great chair for cheap!