The Chomsky Hierarchy is a hiearchy of classes of grammars. It is useful in showing what language a type of grammar can recognise. More powerful grammars contain less powerful ones, and thus can also recognise language recognised by the “inferior” grammars.
From most powerful to least powerful:
Recursively Enumerable (Type 0):

Makes use of unrestricted rules. Both sides of each rule can have any number of terminal and nonterminal symbols.

As expressive as Turing Machines.

A B C > D E
ContextSensitive Grammar (Type 1):

Right side of each rule must have at least as many symbols as the left side.

As expressive as LinearlyBounded Turing Machines.

Can represent languages such as \( a^nb^nc^n \).

A X C > A Y C
ContextFree Grammar (Type 2):

The left side consists of a single nonterminal symbol.

Equivalent to a pushdown (stack) automaton.

Can represent languages such as \( a^nb^n \).
Regular (Type 3):

A nonterminal symbol on the left side, and a terminal (optionally followed by a nonterminal) symbol on the right hand side.

Equivalent to finitestate automata.

Can represent languages such as \( a^*b^* \).