The best definition I can come up with would be an example.
The State of North Carolina issues a billion dollar bond. They now have a long term liability of one billion dollars and the interim interest payments thereon.
For whatever reason they choose to swap those fixed interest payments into floating rate payments and they create a transaction with JP Morgan to do that. Now JP Morgan owes a stream of payments to NC and NC owes a stream of payments to JP Morgan and its bondholders. If JP goes out of business the only effect is to the stream of interest payments owed to NC on a billion dollars NOT the billion dollars itself.
The billion dollars is a notional amount defining the agreement between JP and NC. The other billion, the one NC owes to all of the investors that own its bonds is the real deal.
Similarly, the state of California could issue a billion dollars of debt and in the six months leading up to the issue date they could by a "cap" on interest rates from JP Morgan and that cap represents one billion in notional value. Again, if JP goes out of business the only effect is that California has lost its hedge and the "premium" it paid for protection.
Notional amounts are the amounts being hedged. Derivatives going bust on notional amounts don't necessarily have an affect on the underlying amount of the original issue.
Trillions of dollars of interest rate swaps, caps, floors, etc. are NOT necessarily principal that is at risk.
And depending upon the swap let us say that the stream JPM owed to NC amounts to 20% of the interest that NC owes to the bondholders. Let us suppose further that based on that stream NC took out another bond issue. The taxpayers are now stuck with an extra 20% in interest payments above what the state had budget. Do this a few times and the state is going to have a hard time meeting the payroll for its schools.