A library was a place where books were made available for public use.
Both the Doctor's TARDIS and the Master's TARDIS had libraries. (PROSE: War of the Daleks, All-Consuming Fire, The Dimension Riders, TV: Journey to the Centre of the TARDIS, PROSE: The Quantum Archangel)
The Library of St John the Beheaded was owned by the Catholic Church and located in London. It had a large collection of occult literature and forbidden texts, including lost works by Plato, Aristophanes and Shakespeare, (PROSE: All-Consuming Fire) as well as specific works like the journal of Redvers Fenn-Cooper, (PROSE: All-Consuming Fire) The Gnostic Apocrypha of Nostradamus, (PROSE: Millennial Rites) and Love's Labour's Won. (PROSE: The Empire of Glass)
The Twelfth Doctor, Clara and Rigsy met in front of the Great British Library. (TV: Face the Raven)
The Labyrinth of Kerykeion had the largest collection of human books outside of Earth in the 27th century. (AUDIO: Timeless Passages)
The Library on Kar-Charrat was known as being one of the wonders of the universe in the 42nd century. It contained knowledge from every civilised planet in the universe. Even so, the librarians never liked to advertise its presence and so it was rarely visited and the books were practically untouched. (AUDIO: The Genocide Machine)
The Library, which was built during the 50th century, covered an entire planet and contained "every book ever written". (TV: Silence in the Library)