In the news recently is the story of millions of shoppers having their credit card information stolen when they shopped at Target. The full scope of the information theft is not yet known.
http://www.reuters.com/article/2013/12/20/us-target-breach-idUSBRE9BH1GX20131220The underlying problem faced by shoppers is that when they make a purchase they must give information to the seller: their name, card number, expiration date, etc. Any dishonest person who collects this information is then empowered to make purchases onto their account, since making a purchase just requires the name, card number, expiration date, etc., which were all handed over willingly.
Bitcoin solves this problem through the use of private signatures. To send bitcoins, the transaction from the public address is digitally signed using the private key. This signature is done using an algorithm that allows anybody to verify from the public key that the signature is valid. However, to make additional transactions requires the use of the private key, so merchants (or any other people who see the transaction signature) cannot dishonestly create more transactions.
With bitcoin, you never give anybody the information they need to spend your money like you do every day with credit cards.