In Civilization and Civilization II, the Great Library is a Wonder of the World that allows the civilization that builds it to acquire any technology that any other two civilizations that have acquired automatically. In higher level games, or games with many competing Civilizations, this is perhaps the most important wonder that can be built.
To build the library, the Civilization that is building it needs to have acquired the Civilization Advance of Literacy and somehow rustle up 600 resource shields of production, a lot of gold, or a combination of the two.
In Civilization, the library expires with the advance of the University, and in Civilization II, with the advance of Electricity.