Context-free grammars give a set of rules for generating formal languages. The formal languages generated by a context-free grammar are known as context-free languages.
Context-free grammars (CFGs) were introduced by Noam Chomsky in an attempt to classify formal languages. CFGs are type-2 grammars in the Chomsky hierarchy and generate what is known as context-free languages.
Formally, CFGs consists of a set $V$ of variables, an alphabet $\Sigma$, a set of production rules $R$ and a start symbol $S \in V$.
As an example, the following CFG generates the language of all strings starting with $1$ over the alphabet $\{1,0\}$. The start symbol is $S$.
$$S \rightarrow 1B$$
$$B \rightarrow 1B \; | \; 0B \; | \; 1 \; | \; 0$$
Here the arrow means that the variable on the left-hand side can be replaced by any of the expressions on the right-hand side, where different expressions are separated by a $|$.
To derive the string $1101$ from the previous CFG, we can give the following derivation, starting with the start symbol $S$.
$$S \Rightarrow 1B \Rightarrow 11B \Rightarrow 110B \Rightarrow 1101$$
