The problems it solves are things like horizontal scroll bars when the user makes a web browser narrower, having to "pinch and pan" the screen when viewing on a mobile devices, or not being able to use certain functionality as mobile do not have "hover" functionality.
.net magazine listed this as the number 2 web design trend for 2012 (behind progressive enhancement) and Mashable has called 2013 "The year of responsive web design".
At NBS we try and be innovative and respond quickly to new technology. So responsive design was important to us when we were asked to develop the RIBA Plan of Work 2013 website, which was launched in March this year.
The website looks great on a traditional computer (as you'd expect)...
|On a lap top|
And of course a little shout out to the interface designer who actually did this work - @StylizeRuss (also responsible for the BIMBingo sheets).
|Sketch showing considerations of the fluid grid and CSS rules|