The Ripple network works a lot like the Bitcoin network: units of value can be stored at a wallet address, and whoever owns the wallet's private key can send funds to other wallets. The big difference is that while Ripple does have its own, internal currency (XRP, which right now is used mainly to pay transaction fees to the network), you can also send ”IOUs“ — units of value issued by some third party.
So, some party might take USD (could be your bank, or a third party like PayPal or Stripe), Bitcoin (could be your favorite Bitcoin exchange), Euros, etc., hold them, and issue IOU to your Ripple wallet. So you give $100 USD to PayPal, they send an IOU worth $100 PayPal USD to your Ripple wallet (signed by them and stored in the Ripple blockchain).
You can send all/part of that PayPal-issued $100 to anyone else via Ripple, just like you'd send XRP. Whoever wants to can redeem them for USD at PayPal — you just deposit it into a special Ripple address that PayPal gives you and, boom, the money shows up in your PayPal account (because PayPal was holding it all along).
Here's where it gets even cooler. The Ripple protocol includes an “order book” where you can trade currencies. So, if I'm buying something online and use Euros and the company I'm paying uses USD, my Ripple client could automatically get an IOU for x Deutsche Bank EUR (from my bank), trade it on the Ripple network at the current exchange rate for y Bank of America USD (the seller's bank), and then pay them via Ripple in USD that they can deposit straight into their bank account!
Eventually most transactions might be done in straight XRP so we save on transaction fees (from trading).
That's the plan anyway, as I understand it. Ripple ain't there yet. But, there are companies out there right now that issue IOUs in a few currencies and are trusted enough that people might accept them for payment. So I’m pretty excited.