Cognitive linguistics is a general approach to linguistics that is based on (1) cognitive psychology, particularly schema theory and related studies of how humans perceive and categorize the world; and (2) the role of metaphorical meaning in language. The field includes cognitive semantics, for word meanings and particularly metaphor based semantics, and cognitive grammar for language structure.
Different flavors or varieties of cognitive linguistics exist. Some focus more on metaphor, interpretation, and pragmatics, and some of these approaches tend to overlap with functionalist linguistics. Others focus more on categorization, schemas, and grammatical structures in language, and offer a more structured approach to linguistics. Some cognitivists like Ray Jackendoff combine cognitive linguistics with some worthwhile insights and methodology from constraint-based generative grammar, for a more formal theoretical approach.
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.
The role of metaphor in language use was demonstrated by George Lakoff and others, and metaphor turns out to play an extensive role in language. Oftentimes, words are applied metaphorically in ways that we are not aware of. For example, the word "up" indicates upward position, but in human life it is connected with other concepts, so the word is metaphorically extended to communicate such concepts. We associate "up" with filled containers, and thus, increase in amount, even for non-physical entities like prices (fill up the container, the prices went up), completion of events (let's finish up), degree of completion or degree of action (I'm fed up with this). Linguistic studies have found that metaphorical extension is a highly productive and regular feature of language - not only lexical meaning, but grammatical categories like noun classes and other grammatical categories in different languages.