Noam Chomsky's term for the internal, intensional, and individual knowledge of language in a speaker's head. It is the object of scientific study of language. It contrast with E-language, E for external and extensional, which Chomsky claims does not exist as a coherent or definable object of study.

His claim, the essence of his theory of Universal Grammar (UG) from the very beginning of his work on it in the mid 1950s, is that humans are genetically endowed with a language faculty. The individual child, exposed to a variety of primary linguistic data, develops or 'grows' their inherent language organ, and learns the settings and lexicon that make up the specific language they're growing up amongst. They construct a new I-language, very similar to those of their parents and peers, but probably slightly different in a few respects. Language change is believed to be mainly carried out by children at this stage, reanalysing what they are exposed to.

The I-language is a form of knowledge. It allows speakers to generate and to understand sentences, and also to make judgements of grammaticality. Speakers can make very subtle judgements about degrees of acceptability, as to how a complex sentence fits their own standards of grammatical formation. They might, in addition, have some external standards to bring to bear: such as notions of 'correctness' taught in some cultures, but these are largely irrelevant to their internal grammars. The I-language is largely fixed in early years before schooling can affect it.

A speaker's 'competence' is their ability to produce and understand sentences of their language. This ability comprises, minimally, their own I-language, which is the generative procedure for them; but it also takes in pragmatics about when sentences are appropriate and how to interpret implications and social cues, and learned understanding of other dialects and social registers of language. So competence is not the same as I-language, though it was formerly effectively the technical term for it before Chomsky distinguished them.

Speakers don't always speak perfectly: real linguistic data is full of wrong starts, hesitations, and mistakes. Not all utterances conform to anyone's grammar. Moreover, the child growing up and acquiring its language is exposed to this mass of disparate and misleading data. But the child's UG endowment enables it to construct an I-language, an internal grammar of its own, disregarding the incoherent and generalizing from perceived regularities. This inexact and unreliable repertoire of data is called performance. Various mental faculties handle the processing of performance, including an articulatory production system, a parser to disentangle heard (or seen) syntax and reconstruct it for the generative apparatus, and possibly a module for pragmatics.

Since the I-language does not generate performance, and not all performance is grammatical, there is no reason why the set of all performed utterances should constitute a special set. All sorts of things get said, but those that do are not an especially informative or significant sample; and often, judgements on abstruse and implausible sentences constructed by linguists provide valuable information about I-language even though they're very unlikely to occur naturally in performance.

What Chomsky has criticized from the beginning is the belief that the set of all actually performed sentences, or some subset of fully grammatical ones, that is the E-language in his terms, is of interest to linguistics. There are no E-languages like English, French, or Japanese, defined as sets of possible sentences of English, French, or Japanese. This notion of E-language as an extension, with I-language as merely a mechanism for generating them, is appropriate for the well-formed formulae (WFFs) of a formal language as found in logic or computing. In these, it doesn't matter precisely how they're generated: all that matters is which ones are syntactically correct.

But the human mind/brain (Chomsky's neutral term to avoid irrelevant discussion) is a biological organ with a specific genetically-derived method of creating language. It is this accomplishment, not the output derived from it, that is the serious object of study. Potentially, two different I-languages (instantiated in two different brains) could generate exactly the same outputs (so two English speakers might agree exactly on all their grammaticality judgements); Chomsky leaves this theoretically open, but doubts that it is at all likely. But what we as linguists want to know is what's going on in their brain when they make these judgements, or produce these sentences.