The halting problem, or rather Turing's proof that it's unsolvable, usually doesn't get specified quite completely, certainly not in pop sci articles.
Turing showed that there is no general solution to determine whether a program running on a Turing machine will terminate or not. Seems simple. Until you look at the details.
"General solution" means that you have to write a program that can determine whether ANY program will terminate. "Turing machine" means that the machine has infinite memory and can take infinite time.
So the proof basically says that it is not possible to write a program (that itself has to run on a Turing machine) that can determine whether any possible program, running on an impossible infinite computer, will terminate, given infinite running time.
That doesn't really connect with practical applications very well. It's perfectly possible to prove that a particular program will halt. It's also perfectly possible to determine whether any given program will finish or not in a particular amount of time.
The authors of the paper come up with a dilemma involving keeping a possibly evil programmer in jail. The real world solution is that if you can't prove evil intent within some specified amount of time, you release the programmer.