If you’re reading up on Turing machines, you’ll find the notion of the “halting problem”. In other words, I could ask you the question “can you come up with an algorithm that could take another algorithm as (name removed by moderator)ut and determine whether that algorithm will eventually find a solution and halt on all (name removed by moderator)uts given to it?”.
Turing proved that this problem cannot be solved algorithmically: there is no program which can make this determination for all algorithms and all (name removed by moderator)uts. This isn’t the only
undecidable problem out there, though; there are many others.
I don’t know if this helps, though: it’s not that the halting problem is “solved non-algorithmically”, it’s that it’s “non-solvable algorithmically.” There’s an important difference there, of course.