The tape machine model is one way of looking at it. Another is the type of functions (expressions) that can be computed. A turing complete process can compute g

The tape machine model is one way of looking at it. Another is the type of functions (expressions) that can be computed. A turing complete process can compute general recursive functions. I think of this very informally as having not only for() loops with fixed bounds on the loop index, but also while() loops where you don't know beforehand how many loops it will take to compute something. Alternatively you can express the same thing in terms of recursion. One characteristic is that it's possible to describe computations that never finish. Another is that you cannot always tell from looking at a particular program if it will ever finish. Exactly how or if the result of a computation becomes apparent to the outside world, I think is not part of the theoretical description.