Correspondence between automata and formal grammars?
The expression "one-to-one correspondence" seems a little too strong for me. It suggests that for every grammar there is a specific automaton. It should be read as: for every grammar an automaton can be constructed for the same language and vice-versa.

Context-sensitive languages are accepted by linear bounded automata and generated by context-sensitive grammars.
Context-sensitive grammars have productions of the form $\beta A \gamma \to \beta\alpha\gamma$ where $A$ is a nonterminal and $\alpha$ is nonempty. They are equivalent to length-increasing (more properly noncontracting, or monotone) grammars, which have the form $\alpha\to \beta$ where $|\alpha| \le |\beta|$ (usually $\alpha$ is assumed to include at least one nonterminal).

The languages of Turing machines are generated by unrestricted, or type-0, grammars. See Chomsky Hierarchy.