The halting problem is an example of a problem that a computer cannot solve. On the face of it, it might seem that computers can do anything, given enough resources and time. However, Turing showed that this intuition is false. Later on this was made more tangible by showing that a computer cannot decide whether a given Diophantine equation has solutions in the positive integers.
This is similar to Gödel's incompleteness theorem, which shows that the axiomatic method doesn't provide an answer to all possible questions. In fact, no finite list of axioms is enough to determine the truth of every statement regarding the natural numbers. Later on, a natural example emerged in the realm of set theory, namely the continuum hypothesis.