s or machine
s are those
that can express
s, that is, they can emulate arbitrary Turing machine
This is not at all the same as the capability to solve any mathematical problem: many perfectly defined mathematical problems are provably unsolvable.
As a matter of fact, the Turing machine was invented to prove this.
A simple Turing-complete formalism is, obviously, the Turing machine; another, the lambda calculus; another is the arbitrary rewrite grammar often used as the last formalism in the Chomsky hierarchy.
Very small subsets of most programming languages are already Turing-complete, on the condition that they can access arbitrary amounts of memory. C, for instance, isn't, because it must address all memory with pointers of a fixed size; but a variant that uses pointers of arbitrary size would be trivial to define.