Generative Grammar is thought to be (an abstract) ``device that generates all the grammatical sequences of (a language) and none of the ungrammatical ones'' (Chomsky, Syntactic Structures, 1957). The best known of the generative Grammars is the transformational Grammar of (Harris/) Chomsky. In later years though Chomsy has moved away from the transformational view.
The sense in which ``Grammar'' is used above is not in the common sense of ``syntax'', but to refer to the whole of the systematic description of language, including phonology, semantics and syntax.
Zellig Harris, 1950s (was a ``Bloomfieldian'') His work in syntactic analysis started the process that overturned the domination of descriptive linguistics. Harris used transformations to relate complex sentences to simpler kernel sentences. However Harris did not want to tie the transformations into a single theoretical framework or to posit an abstract level of analysis, something that his student Noam Chomsky did.
Noam Chomsky Ph.D. thesis, 1955, published as Syntactic Structures, 1957; Review of Skinner's book in 1959; Aspects of the Theory of Syntax, 1965.
SS, a book of less than 120 pages, revolutionised the study of linguistics. In it, Chomsky stressed the creativity or ``open-endedness'' of human language: the ability of native speakers to produce and understand sentences never heard before. The majority of the sentences produced were ``new''. Though earlier linguistic theorists including de Saussure had assumed this, it had been neglected by the Bloomfieldians, perhaps due to their agenda of stressing the descriptive as opposed to the prescriptive aspects of language.
In SS Chomsky drew a distinction between the sentences generated by the grammar (the language) and a sample of utterances produced by native speakers (the corpus). (The latter cannot by itself yield the grammar.) Thus Chomsky rejected the ``discovery procedures'' sought by the Bloomfieldians: ``a linguistic theory should not be identified with a manual of useful procedures.'' Intuition, guesswork, past experience etc. is useful (Chomsky's early empiricist formulation).
The basic elements of a language cannot be discovered independently of the rules for combining the elements. These rules are intuitively known to native speakers: Competence and Performance (his later rationalist formulation). In his later work Chomsky included the intuitions of native speakers as part of the data to be accounted for by the Grammar.