An automaton, is a logical machine that models the process of computation.
An automaton can also be used as a powerful control structure, transcending the block-hierarchical paradigm of structured programming. The following pages are related:
The need for such power is most often felt when parsing or doing IPC.
AM 2010-07-19: 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. A simple program with complex behaviour is one example.