State machines are a generic term. You will find here the different kinds of state machines, with varying expressive power that are used in computer science and actuak programming: https://github.com/achou11/state-machines

In any case, keep in mind that Turing machines are (extended) state machines (albeit with infinite memory) so the expressive power of state machine is at least anything computable (sequentially - turing machine is a model of sequential computation).