Casey Newton, writing for The Verge:

Here’s how Account Key works. You link your Yahoo account to a mobile device, most likely a smartphone. When you go to log in to Yahoo Mail on the web, enter your email address and tab down to the password. Yahoo will recognize that the account has the feature enabled and send a push notification to your device. You can approve or deny the login directly from the notification.

This is a clever way of dealing with the problems associated with password-based authentication.