The name goes back to the 13th century, to the medieval courts of the ancient principality of Catalonia and the kingdom of Valencia respectively, that were both abolished in 1715 by the Decreto de Nueva Planta by Philip V of Spain.
Information about the Generalitat de Catalunya.
The Valencian Generalitat was "restored" in 1982 and these have been its presidents: Joan Lerma, Eduardo Zaplana (through 2002).
Catalan Generalitat: http://www.gencat.net