The way an interest rate swap works is as follow:
I want to bet on the LIBOR interest rate going up from its current position. You want to bet on it going down. So we take out loans against each other for 1000 dollars - you pay, say, the LIBOR rate, and I pay 2% (or whatever the LIBOR rate currently is). This swap has a NOTIONAL value of 2000 dollars, since we each owe each other 2000 dollars. This is what's often referred to when people talk about 700 trillion dollars in derivatives. In reality, at the end of one year you're going to owe me maybe 1022 dollars and I'm going to owe you 1020 dollars, so you're going to end up just paying me 2 dollars.