Here is one way to carve up abstract (mathematical) automata into different classes of complexity (from low to high).

• Combinational Logic
• Finite-state Machine
• Pushdown Automaton
• Turing Machine

There is a somewhat different and older one that is based on the Chomsky Hierarchy (where ND stands for non-deterministic), and the associated language or grammar that they recognize:

• Finite-state Machine (Regular Languages)
• Non-deterministic Pushdown Automaton (Context-free Languages)
• Linear-bounded Non-deterministic Turing Machine (Context-sensitive Languages)
• Turing Machine (Recursively Enumerable Languages)

Further Reading:

https://en.wikipedia.org/wiki/Automata_theory

https://en.wikipedia.org/wiki/Chomsky_hierarchy

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