A responsive design means writing one lot of code that means your website will adapt beautifully on every screen size, device and device orientation. For example, a three-column website layout on an iPad might become a two-column layout when the iPad is rotated, and a one-column width when viewed on a smartphone screen.
Responsive Design is one that adapts to the user's device and, in an ideal world, the user's context so that it displays the content required in the most appropriate and accessible manner, regardless of what kind of web-connected device is being used to view it.
In practice, this means a web page will re-paginated itself as the screen size reduces or increases, displaying in multiple columns when viewed on a desktop computer, but only a single column when viewed on a smartphone.
The concept of responsive design is about much more than simply reformatting content though.