First, let's consider how an internet payment works today.
Ripple has a key distinction: it moves the balances to a decentralized network of computers, controlled not by any single institution, but by an open protocol - like the web.
Bob is now using Ripple. He choses a payment provider that operates on the Ripple network: Bitstamp. Bob gives money to Bitstamp. In exchange, they give him a balance over that amount.
This time, his balance is stored on the Ripple network, decentralized and cryptographically secured. The network ensures that balances are safe and can only be modified by their owners. Even Bitstamp cannot access Bob's balance.
Bob can now sends money to Alice. Note that Alice never signed up with Bitstamp, and has no business relationship them. She nevertheless can choose to receive Bob's money, and the Ripple network will credit her with a Bitstamp balance. Bitstamp cannot oppose this transaction; Bob and Alice may spend their balances as they wish.
There are many other payment providers on the Ripple network.
Brent is a customer of Justcoin, and his Justcoin balance is stored on the Ripple network alongside the Bitstamp balances of Alice and Bob.
Brent also wants to buy from Alice. However, Alice does not accept his Justcoin balance. She does not want to sign up with Justcoin to withdraw her money.
Bob does a lot of business with other Justcoin customers, and so he wouldn't mind a Justcoin balance. Because Bob agreed to this, the network will automatically transfer some of Bob's Bitstamp balance to Alice, in exchange for an equivalent Justcoin balance from Brent.
Bob acted as an intermediary and facilitated a payment between Brent and Alice, and two different payment providers. He can choose to charge a small fee for this service.
Because customers from different providers constantly want to send money between each other, there are many people like Bob on the network, and the market will keep the exchange fee low.
Bob again needs to pay Carol in Euros. The network will convert his USD balance into a EUR balance using the best exchange rate available on the market.
Bob could even pay with Bitcoin, his frequent flyer miles, or gift certificates. As long as someone on the network is willing to accept his currency, he'll be able to have them exchanged, automatically during the payment, for the Euros that Carol is asking for.