The halting problem is "undecidable": there's no algorithm which can tell you if an arbitrary program with an arbitrary input will halt or not. IE, if you give me a proposed "halt-detector", I can inspect it and come up with a program and input for which it would give the wrong answer.
Most people think that's a cool mathematical theory but don't know the real world relevance, because computers aren't Turing machines and have finite memory, so all programs are guaranteed to halt. I once saw a keynote speaker say their programming language was stronger than Turing-complete because all programs in it had a timeout, and so they solved the halting problem.
To better understand why the halting problem is so important, it's better to look at what the world would be like if it was solvable.
A recursively enumerable problem is a boolean problem where any true answer can be determined in a finite time. We can come up with an algorithm that, if the answer is "true", halts and returns "true", and if the answer is "false", either returns "false" or runs forever. Here's a recursively enumerable problem: