The automaton, which is sometimes used interchangably with machine, is one of the basic models of computation:
It can also be used as a powerful control structure, transcending the block-hierarchical paradigm of structured programming:
The need for such power is most often felt when parsing or doing IPC.
AM (19 july 2010) Quite apart from the above, automata are also studied as mathematical objects - see the book A New kind of Science by Stephen Wolfram for instance (http://www.wolframscience.com/thebook.html ). Here is one example: A simple program with complex behaviour