Turing machines are a big deal because when you change the definition of a Turing machine (by letting it use arbitrarily many symbols, or giving it multiple tapes or a multi-dimensional tape, or letting it operate nondeterministically, or making its tape finite in one direction...) it usually can still solve exactly the same set of problems, which strongly suggests that Turing completeness is a "natural concept". A lot of computational systems are Turing-complete, and all of the discrete computational systems we've built are no more powerful than Turing machines. Physical reality might also be Turing-computable (Church-Turing-Deutsch principle), though we don't know that for sure (for instance, physics might involve infinite-precision calculations on non-computable numbers).