Clearest article I've ever read on it.
Is God more powerful than a Turing machine?
The Church-Turing Thesis proposes there is a legitimate concept of "most powerful computational model that can perform, at most, a finite number of calculations in a finite period of time", and that the Turing machines (and every equivalent computational model like modern logic, math, most computer programming languages, etc.) are that model. This is not proven, which is why it's "only" a thesis.
When Penrose suggested (incorrectly IMHO) that there were tiling problems that humans could solve that a Turing machine could not, this was a fascinating development, because it suggested there either was something infinite in the "computation" the human mind performed, or it was still finite, but there was (at least one) layer greater than a Turing machine that was still finite in nature, and hence the Church-Turing thesis was false. Either development would have been fascinating. So, too, would "quantum computing", should that pan out, because that suggests one could do an infininte number of computations in a finite period of time, solving huge almost-prime factorizations, etc. This in turn would demonstrate the universe itself was on a higher level than a finite computational device.
But it's still all speculation at this time. The Turing machine is still "it" as far as anyone has demonstrated.