Responsive web design is the ability of website content -- text, information, images, etc -- to adapt fluidly to a user's viewing device.
The idea is that regardless of the device -- computer, tablet, phone -- the design needs to automatically fit the screen size and be easily readable.
In the past some sites would look attractive on a computer screen, but when visited by a tablet or phone they would force the visitor to zoom in and out of the site to be able to read or click around. This would severely hamper the viewing experience and reduce the likelihood of a visitor to return, use or shop on the site.
Designers soon began to find ways to simplify designs so they would look the same or good on all devices. Eventually coding methods were devised that could shift the look of a site based on the size of the browser window, such as separating the design and information into blocks and shifting these blocks up or down, left or right, depending on the screen size.
Most newer sites built on content management systems (CMS) such as WordPress now have the ability to recognize a visitor's browser or device and adapt themselves accordingly. They either can resize the look of the site by shifting the content around or by redirecting the visitor automatically to a secondary mobile site. Most WordPress themes have the ability build in and most WordPress designers plan ahead with responsiveness in mind when building sites.
Google has recognized the importance of responsive design and made the ability of a site to be responsive a priority in their ranking.