Cats actually have a long and rich history of being associated with wisdom, study and books in general, and libraries with resident cats are not as uncommon as you might think. It's likely that their original role was that of pest control specialist, but proponents of library cats believe that cats increase patron attendance and overall circulation, and help keep the library a vital and loved component of its community.
Obviously, I'm also in favor of the idea. This map here is a handy tool in understanding where you might be able to see a library cat for yourself (it includes statues, stuffed animal mascots, and ghost cats, so read carefully to avoid potential disappointment).
Here are a few of the more well-known library cats from the recent past and also the present:
Dewey Readmore Books
The resident cat of the library of Spencer, Iowa starting in 1988 until his death in 2006, Dewey greeted patrons, attended staff meetings, and conducted in-depth inspections of any box he could find. You can learn more about Dewey by reading the book that librarian Vicki Myron wrote about her dear friend. I must warn you though - I've read it. It made me cry.
Browser is a library cat who has embraced the modernity of social media as well. He blogs here, and this is his Flickr account. Browser likes to spend time with the kids during story time, and fall asleep in the reference section.
Browser isn't the only library cat with a blog. Pages has one, too, and she uses it to promote events at the Valley Center Public Library (and occasionally write about zombies). Though not everyone was in favor of allowing Pages to stay at the library, it is now her home, and she likes to jump from shelf to shelf when she's not busy with her blog.
Sir Eli is a gentleman cat who works part-time at the Los Robles Elementary School Library. Twice a week he helps kids with their reading, and is always available for comforting those who need it - when he's not at the library, Sir Eli accompanies librarian Marily Barsileau on visits to the hospital to be his fluffy self around sick patients.