You can't work long in computer science until you have to understand the halting problem. The halting problem is concerned with determining whether a program will finish running or continue to run forever. It has a storied history. In 1936, Alan Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. His proof was made famous since in this paper, he provided a mathematical definition of a computer and program, which became known as a Turing machine and the halting problem is undecidable over Turing machines and is one of the first examples of a decision problem.
Programmers generally try to avoid infinite loops—they want every subroutine to finish (halt). In particular, in hard real-time computing, programmers attempt to write subroutines that are not only guaranteed to finish (halt), but are guaranteed to finish before the given deadline.
Moreover, the Halting problem lets us reason about the relative difficulty of algorithms. It lets us know that, there are some algorithms that don't exist, that sometimes, all we can do is guess at a problem, and never know if we've solved it. If we didn't have the halting problem, we would still be searching for Hilbert's magical algorithm which inputs theorems and outputs whether they're true or not. Now we know we can stop looking, and we can put our efforts into finding heuristics and second-best methods for solving these problems.