A Turing-complete language is capable of functionally expressing a Turing machine. That is, given the theoretical construct, can the language encode it? An equivalent framework is lambda calculus. Any Turing machine can be expressed as lambda-expressions and vice-versa.
The physical world analog to the Turing machine is the Universal Constructor, a machine that can construct a duplicate of itself or a functional analog of any other machine that can duplicate itself, given enough time and raw materials.
These concepts are relevant because it's not clear if there is a class of problems not solvable by Turing machines (but solvable by another class of machines). As far as we know, human beings themselves are Turing machines (and constructors) and nothing more.