| Language | Machine |
|---|---|
| Regular language | Deterministic or nondeterministic finite-state acceptor |
| Deterministic context-free language | Deterministic pushdown automaton |
| Context-free language | Nondeterministic pushdown automaton |
| Context-sensitive language | Linear-bounded automaton |
| Recursive language | Turing machine that halts |
| Recursively enumerable language | Turing machine |
Not all language classes fit neatly into a hierarchy. For example, we have discussed the linear languages, which (like deterministic context-free languages) fit neatly between the regular languages and the context-free languages; however, there are languages that are linear but not deterministic context-free, and there are languages that are deterministic context-free but not linear.
In fact, mathematicians have defined dozens, maybe hundreds, of different classes of languages, and write papers on how these relate to one another. You should know at least the four "classic" categories that are taught in almost every textbook on the subject.