This may have been asked here at one point but couldn't find anything when searching.  I looked up the re-release of the book Brideshead Revisited at Amazon and it's binding it stated as being "cloth hardcover."  Can someone enlighten me as to what that actually is?

It means the binding is made of fabric rather than cardboard or heavy paper stock. Most older hardcovers are made of "cloth" - most newer ones are made of something similar to cardboard unless they are very good quality.

More clearly, Bren, don't you mean that it is "cloth covered boards" rather than just inked or plain boards. Maybe that's not what you mean, but isn't there cardboards underneath the cloth? Otherwise, what would make it hard?

I've got a copy of The Divine Comedy that's a cloth cover.  I don't know what they used to make for the hard part, but the cloth is very obviously cloth.  It's kind of like those spandex book covers you can get for schoolbooks and the like, that stretch over the cover, although obviously not as stretchy (it's just a little loose).  I'm sure I have some that are cloth but tight and it's easy to see how they replaced it with textured board, as I couldn't easily identify which is which.  At least with that one, though, it's obvious, and it makes the book's age and quality stand out.