The halting problem:

Given a program and an input, determine whether that program, given that input, will ever halt, or if it will loop infinitely.

One of the fundamentals of academic computer science is learning that it is mathematically impossible to solve the halting problem - which is why the grandparent comment simply notes "the halting problem" with no further explanation; it's a famous problem. (Don't feel bad, everyone has to learn something the first time!)

The poem linked elsewhere in this comment tree is a simple and pithy description of the mathematical proof.

For (many) more details, Wikipedia has plenty of information.