What is ths author talking about?

For instance, we know that a Turing computer cannot calculate whether all such computers will terminate.

It cannot calculate whether two such computers are equivalent (performing an equivalent calculation).

And what exactly "artificial computation", as a class? In contrast to what, and what is the difference?

There is something called the Chuch-Turing Thesis; it is an unproven conjecture which says that any function that can be computed by an 'effective method' can also be computed by a Turing machine. No counterexample has been found.