Cognitive grammar, or the more general domain of cognitive linguistics, is a general approach to analyzing linguistic structure that is based on cognitive psychology, particularly schema theory and related studies of how humans perceive and categorize the world. It is one of several major groups of linguistic theories that are popular in the linguistics community.
This paradigm began with the insights of Gestalt psychology, and later, schema theory in cognitive psychology. It assumes that language is a cognitive domain that is grounded in cognitive psychology and other mental faculties, and that language must have arisen from cognitive faculties rather than just on its own, and hence, language is crucially connected with cognition - be it natural semantic categories in our world, or human social cognition. It combines these psychological insights with the key insights from the study of metaphor - that metaphor is a key element of meaning in language, and hence, a whole field of cognitive semantics exists for exploring this.
Construction Grammar, in particular, takes a principled, structured theoretical approach (but not so formalized as Jackendoffian linguistics), focusing on the meanings associated with various types of sentence structures. These structure-meaning relationships are grounded in schema theory and the metaphorical extension of meaning. For example, "give me the ball" is a standard dative sentence (focusing on the recipient or the one benefiting from the action), while "give the ball to me" is has a transfer-of-possession meaning. These structures are then applied to other motion verbs like "throw / toss me the ball" or "throw/toss the ball to me" or "hurl the ball to me" (but not *"hurl me the ball," where a dative/benefactive meaning would be awkward with this verb).
Because it is grounded in a practical understanding of metaphor and practical insights from cognitive psychology, cognitive grammar lends itself to classroom language pedagogy, particularly in informing teachers of linguistic principles that can be readily conveyed to students. This includes the metaphorical meanings of phrasal verbs, metaphor in much of the vocabulary of a second language, and the fact that grammatical forms and structures are not just structures to be learned, but these structures themselves have an inherent meaning that should be conveyed to students. Examples include many structures and forms that students may struggle with - verb tenses, especially the perfect tense; definite / indefinite articles; phrasal verbs; prepositions; various sentence structures such as passives, different types of intransitives, transitive sentences, and others. These in turn could be readily adapted to forming more intuitive explanations for students, which could be combined with inductive learning exercises and communicative language activities.