Public key cryptography is probably the most significant cryptographic advance since the discovery of the monoalphabetic substitution cipher thousands of years ago. In short, it provides an elegant solution to the problem of key distribution. Normally, two people wishing to exchange encrypted messages must exchange both the message and the key to decrypt it. Sending both over an insecure connection is obviously unsafe and, if you have a safe connection, there is little need for encryption. Based on some fancy math, public key encryption systems let Person A encrypt messages for Person B using only information that Person B can make publicly available (a public key, like mine).
Now, quantum computers running Shor’s algorithm threaten to ruin the party. Two groups claim to have achieved some success. If they manage the trick, the consequences will be very significant, and not just for PGP-using privacy junkies. Public key encryption is also the basis for all the ‘https’ websites where we so happily shop with credit cards. If a fellow in a van outside can sniff the traffic from your wireless network and later decrypt it, buying stuff from eBay and Amazon suddenly becomes a lot less appealing.
Thankfully, quantum computers continue to prove very difficult to build. Of course, some well-funded and sophisticated organization may have been quietly using them for years. After all, the critical WWII codebreaking word at Bletchley Park was only made known publicly 30 years after the war.
For those who want to learn more, I very much recommend Simon Singh’s The Code Book.