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Let me add to what ntece has answered,

What is automaton ?

A machine which performs a range of functions according to a predetermined set of coded instructions.

Yes, we have real life examples like vending machines or traffic lights.They all have states here they are finite states.

Good Reference on FSM (Finite State Machine)

http://faculty.etsu.edu/tarnoff/ntes1900/FSM.pdf

Automata Theory is a branch of computer science that deals with designing abstract self propelled computing devices that follow a predetermined sequence of operations automatically. An automaton with a finite number of states is called a Finite Automaton.