Uncut books refers to the now probably obsolete practice of books having pages with untrimmed edges which also bind them together. These must be cut before the pages can be seperated, and the book can be read. The fact that the books' pages have not been cut shows Gatsby has no intention of reading them; they are just for show.
The books are contained in a 'high Gothic library, panelled with carved English Oak'; the attention to detail is impressive but these details suggest an aristocratic background which Gatsby does not possess. It is just for show just like the books. The whole thing is a sham.
Gatsby's saving grace is that the books and the library are not to show off to everybody - just Daisy. They, like the wealth which has bought them, are merely a means to an end: his dream of winning Daisy back.
So the books symbolize Gatsby's vision of himself and his dream but also the fact that they lack true depth. They're fragile also: in a telling detail, the owl-eyed man who presides over Gatsby's library, shoves back one of the books 'muttering that if one brick was removed the whole library was liable to collapse'. This persona, this dream self Gatsby has constructed, is likewise liable to collapse at any time.
It is in Chapter Three of The Great Gatsby that Nick meets Owl Eyes and discovers the uncut books in Gatsby's library. The uncut books are significant because they prove that Gatsby has never read any of these books since the pages are still folded together and can only be read if they are 'cut' apart.
The uncut pages are a potent symbol of the fact that Gatsby is not who he claims to be. In fact, the uncut books are part of his great façade: he wants people to believe that he is cultured and sophisticated and he cultivates his image accordingly.
Just like his lavish parties, Gatsby's uncut books demonstrate that there is more to his character than meets the eye. As the reader later learns, the reputation which he has so carefully cultivated has one objective: to win back Daisy Buchanan. As such, the fact that he would display uncut books demonstrates the strength of this determination.