Noam Chomsky

The formalism of context-free grammars was developed in the mid-1950s by Noam Chomsky, and also their classification as a special type of formal grammar (which he called phrase-structure grammars).

A context-free grammar provides a simple and mathematically precise mechanism for describing the methods by which phrases in some natural language are built from smaller blocks, capturing the "block structure" of sentences in a natural way.

In Chomsky's generative grammar framework, the syntax of natural language was described by a context-free rules combined with transformation rules. In later work (e.g. Chomsky 1981), the idea of formulating a grammar consisting of explicit rewrite rules was abandoned. In other generative frameworks, e.g. Generalized Phrase Structure Grammar (Gazdar et al. 1985), context-free grammars were taken to be the mechanism for the entire syntax, eliminating transformations.

A formal grammar (sometimes simply called a grammar) is a set of formation rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form. 

Formal language theory, the discipline which studies formal grammars and languages, is a branch of applied mathematics. Its applications are found in theoretical computer science, theoretical linguistics, formal semantics, mathematical logic, and other areas.