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Such a machine can be given input, which is either rejected, or accepted. It's like a vending machine. When something is bought, coins (or money) needs to be inserted into the machine. If these are the right coins, they are accepted, and the requested item is dropped so it can be removed. If the coins are wrong, they are rejected.
Internally, the automaton has different states it can be in. Feeding it input may (or may not) change its state. That way, the automaton goes through all the input, consuming one item (which mathematicians call a symbol) at a time. When no symbol is left, the automaton is in a certain state. This may be an final state. In this case the input is accepted. Otherwise, the input is rejected.
If the machine has a countable, finite number of states, it is called finite state machine. A diagram that shows all the states, and transitions of such a machine is called finite state diagram.
Like in real life, there are machines that are too complex to understand. The mathematician and computer scientists therefore ask themselves if a certain automaton is minimal. If it is not minimal, there must be another automaton with fewer states that can do the same thing. An example of an automaton is the turing machine.