From the Wikipedia page 
A national library is specifically established by the government of a nation to serve as the pre-eminent repository of information for that country. Unlike public libraries, they rarely allow citizens to borrow books. Often, they include numerous rare, valuable, or significant works; such as the Gutenberg Bible. National libraries are usually notable for their size, compared to that of other libraries in the same country. Some national libraries may be thematic or specialized in some specific domains, beside or in replacement of the 'main' national library.
Some national entities which are not independent but who wish to preserve their particular culture, have established a national library with all the attributes of such institutions, such as legal deposit.
Many national libraries cooperate within the National Libraries Section of the International Federation of Library Associations and Institutions (IFLA) to discuss their common tasks, define and promote common standards and carry out projects helping them to fulfil their duties. National libraries of Europe participate in The European Library. This is a service of The Conference of European National Librarians (CENL).
The National Libraries of Africa webpage is .
The European Board of National Archivists listing is at .