The short text at the back of the book/cover that is usually a summary of what the book is about, I believe. Also often had some short reviews, like the op title refers to. (The stuff you look at when you are deciding if you want to read the book or not, generally.)
In one sense, each such quote is a blurb*. It's not clear to me whether "Klappentext" refers to everything on the back of a book, but if it does then blurbs form all or part of that.
In another sense, we might use "blurb" to mean everything on the back of a book. I'm not completely sure which is correct, but when in the plural for one book, it's definitely being used as a synonym for "quote" with a little added context.
* I'm not even sure whether "blurb" refers to only a quote or to the quote and its following attribution. Dictionaries are not clear on this.