Cognitive Linguistics defines itself against the background of standard, or 'formal' linguistics. While there are a number of different schools of formal linguistics, the most commonly taught is Generative (Principles and Parameters) Linguistics. According to Generative Linguistic theory, a sentence is a string of symbols over some alphabet, and a language is a set of sentences. A grammar is a finite set of rules that uniquely defines the (infinite) set that is a language. Commentary
The symbols in this theory are essentially meaningless. The study of Semantics (meaning) is secondary to and entirely dependent on the study of syntax (linguistic form). Syntax is thought to be autonomous; it interfaces with Semantics but is a distinct thing in its own right; there is a distinct 'syntax module' in the brain.
Insofar as semantics is considered, it is defined in terms of truth-conditions. That is, the meaning of a sentence is the set of world-states in which it is true. Meaning has nothing to do with human concepts; more generally, language does not necessarily have anything to do with people, or with communication.
The best-known proponents of this theory (or rather, set of theories) are Noam Chomsky and Steven Pinker.
There are a number of other formal theories, including LFG and the Phrase Structure Grammar family, the best-known of which is HPSG.
The theoretical commitments of Cognitive Linguistics are incompatible with all these theories, but they are most markedly and fundamentally incompatible with Principles and Parameters theory.
Cognitive Linguistics is still something of a fledgeling field. Important work in what is now Cognitive Linguistics had been going on since the late '70s, but The first book to pull together the different aspects of the field into something resembling a unified framework was Women, Fire and Dangerous things: what categories reveal about the mind (Lakoff 1987).
This puts the field at something between 14 and 25 years of age. What I personally find amazing is how much has been accomplished in that short time.
Cognitive Linguistics is a fundamentally Empiricist enterprise. As such, we think the data are more important than any theoretical mechanism. The theories exist to explain the data. They do not exist to restrict which data we look at.
Of course, this doesn't stop us from getting caught up in theories, but we try not to.
Cognitive Linguitics is not a sub-field, the way the study of Syntax or Phonetics are sub-fields. It is a theoretical framework, a perspective on the whole of langauge. Thus a Cognitive Linguist's specialization may be Phonology or Semantics or any other sub-field of Linguistics.