As Blizzard plans to go live with the Real Money Auction house (RMAH) for Diablo III, the publisher has announced that using the service will require a Battle.net Authenticator for security.
It's no surprise that Blizzard is making security a top priority with the RMAH, as real money transactions will be conducted through the service.
There are multiple ways to acquire the authenticator, one of which is to download a smartphone application which you can run when the game asks you to input a numerical code whenever you start up the game. If you don't have a smartphone, you can opt to purchase a keychain for $6.50 or use the dial-in service to call for a code.
The authenticator will remain an optional component to playing Diablo III, but you should get one anyway if you want to keep your account secure. Given that it is more or less free, there's no reason you shouldn't be using it.