Responsive design provides a way for your site's content to be viewed in an optimal way independent of screen size. It subscribes to the "one web" idea where you only build one site for all channels, instead of a dedicated mobile version.
Responsive design is a good solution for many web site owners and builders, and while it won't suit every use case it's certainly a compelling and popular concept.
Responsive design uses a blend of fluid layouts, CSS3 media queries and flexible images that when combined allow your site to adapt to the device it's being viewed on.
Your standard desktop site might be three columns, but in your average smartphone you might want to display only one column - this is entirely doable using responsive design - it's a way of re-flowing the content and layout to suit the width of the device.