A Turing-complete programming language just means that you can compute anything that can be computed using that language. In other words, it can be used to writ

A Turing-complete programming language just means that you can compute anything that can be computed using that language. In other words, it can be used to write arbitrary computations. You don't need much to achieve this; off the top of my head, I think that being able to read and write to storage, basic addition, conditionals and loops are enough. (Formally, it means that your language can be used to simulate a single-taped Turing machine.) A non-Turing-complete language is any language which does not have one of those capabilities. Because the bar is so low, most languages you have used in your life are Turing-complete. For example, you can imagine a simple calculator-like language for just writing mathematical expressions. It can do all basic math operations, has conditions, but no ability to repeat things (no loops or recursion). That language is not Turing-complete. You can trivially prove that programs from such a language will finish - they can't not! there's no ability to impede progress.