Official definition from Jancis Robinson’s The Oxford Companion to Wine:
Noble rot, also known as pourriture noble in French, Edelfaule in German, muffa in Italian, and sometimes simply as botrytis, is the benevolent form of botrytis bunch rot, in which the Botrytis cinerea fungus attacks ripe, undamaged white wine grapes and, given the right weather, can result in extremely sweet grapes which may look disgusting but have undergone such a complex transformation that they are capable of producing probably the world’s finest, and certainly the longest-living, sweet wines. Indeed, the defining factor of a great vintage for sweet white wine in areas specializing in its production is the incidence of noble rot.