In a great answer, a community member gave the following proof sketch that the halting problem is undecidable:
Proof that the halting problem is undecidable. If there were a computable procedure to reliably determine whether a given program/input halts, then design a new program q that on input p first asks whether p halts on input p, and then performs the opposite behavior itself. It now follows that q halts on input q if and only if it doesn't, a contradiction.
Following Mahmud's helpful link below, I found a wonderful article by what appears to be the same user on this subject. In addition, Wikipedia's entry on the problem was helpful in wrapping my head around this. (I also came across this amusing poem about the problem.) I think I understand the way in which this gives us 'unsolvable problems' for any Turing Machine and how a weak form of Godel's Incompleteness Theorem can be derived from the proof that the halting problem is undecidable.
What other, perhaps more general, implications of the undecidability of the halting problem might there be? (In particular, how if at all are these distinct from those of Godel's Incompleteness Theorem?)