That would be called "the Publisher's Logo." Perhaps you are referring to the material which describes who designed the book; the publisher's data; the materials used in its production [etc]. This is called "the colophon."
Added: "Publisher's Mark" is an alternative for "Logo"
Also known as 'onion skin' or 'onion paper'. https://en.wikipedia.org/wiki/Onionskin.
Onionskin or onion skin is a thin, light-weight, strong, often translucent paper. Not made from onions, it is named for their thin, papery skins which it superficially resembles. It was usually used with carbon paper for typing duplicates in a typewriter, for permanent records where low bulk was important, or for airmail correspondence. ~ http://cool.conservation-us.org/don/dt/dt2375.html
Onionskin may be an expression used in book publishing in parts of the USA, as pointed out by EL&U member Eric Hauenstein. The term more frequently encountered in the UK is 'India paper', originally suggested here by EL&U member Brian Hooper. India paper is traditionally a thin light paper produced in Asia or in the West in imitation of Asian thin papers. See: https://en.wikipedia.org/wiki/India_paper
This image (below), of the Deluxe (India Paper) single volume edition of Tolkien's Lord of the Rings (http://www.tolkienlibrary.com/tolkien-book-store/NH0013.htm) gives some idea of how translucent India paper is when used in bound books:
This image (below) is of 'Bible Paper' (referenced by EL&U member mikeagg in his answer to this question) in Schuyler Publisher's Quentel Bible. Bible Paper is essentially the same product as India Paper, but has become the standard term for this sort of very thin paper when used in the production of Bibles. Note that at 45gsm, Shuyler claim that this is the most opaque Bible Paper in the market. (https://tresses.wordpress.com/2014/02/17/schuyler-quentel-nasb-review/)
Just for completeness, here (below) is an image showing onionskin paper. Note the distinctive colour. In former times single plain onionskin sheets were interleaved in books to protect images, and also to prevent heavily inked pages containing images adhering to the page opposite. The drying process applied to onionskin paper gives it a very slightly uneven surface that prevents pages sticking together.
Some Ngrams from Google give us some sense of what is going on. Note that simply searching for onionskin paper (or any of its variant spellings) will find matches for typing and writing paper. This question is asking for information about paper used in book publishing, hence the Ngram search should be for something like 'printed on...', as in this example:
Another Ngram makes the point about the dominance of India paper in publishing even more forcefully:
Finally Onionskin (and its variants) up against India paper (case insensitive):
On a final - final - note, there is some confusion being generated in this discussion over what is meant by the term India paper. This explanation from the ABE Book Trading site should help clear matters up:
Best Answer
The coloured edge-pages are called, unsurprisingly, edge painting or fore-edge painting: that's actually already come up on a Stack Exchange site: Why do some books have colors on the edges of the pages?
Print Ninja corroborate that, and even call the speckled effect you see on some books fore-edge splatter. There are even people who do remarkably detailed work.
An edge mark which defines a particular section is called an edge index. This is produced slightly differently (by printing on the face of the page, at the edge, rather than on the closed edge) but has a similar effect.