Language: http://en.wikipedia.org/wiki/Formal_language

Decision problem: http://en.wikipedia.org/wiki/Decision_problem

Decider: http://en.wikipedia.org/wiki/Machine_that_always_halts

These are fairly foundational concepts when talking about computational complexity.

> However, if I can choose a large enough regex, I can construct a decider for a given finite language. How then are regexes less powerful than C?

Ok so upon reflection I think I see what you are getting at. Practical computers are have limited memory and so are, in principle, equivalent to a finite state machine.

So anything a practical computer can do a, perhaps ridiculously large, regular expression can also do?

Is that close?