I think the problem isn't that Bitcoin can't be explained to normal people, it's that so far we've done a bad job at it. Bitcoin is certainly much easier to understand than PayPal, for example.

All you need to know is that you have a wallet on your PC (or in the cloud) that holds your coins, just like a real wallet, and that you can give out your public address to receive money, or send money to someone else's address. Ideally all of the new addresses and transaction details should always be handled by the client software so that people don't have to deal with that stuff.