I know that Turing-completeness is the ability to simulate a Turing machine, and from what I've read, the reason why we should care about Turing-completeness is that it demonstrates that a machine can "run any program" or "compute the result of any algorithm." While I understand these descriptions, I'm wondering if there's a more mathematically rigorous description of this significance. At a fundamental level, what exactly defines an "algorithm" or "program" or "instruction"? If Turing-completeness is a measure of a machine's ability to manipulate data, what do we mean by "manipulate" and "data"?
The mathematical significance of the concept of Turing machines is to give one possible rigorous mathematical definition of concepts such as "algorithm", "program", "instruction" and "data". It is a very satisfying definition which is known to be equivalent in computational power to many others.
It should be said that there are of course other approaches, which refine and further develop these concepts. For example, in programming language theory operational semantics gives a good account of "instruction" and "computation step". But Turing was the first one.