In a nutshell, responsive web design makes your web page look good and fully functional on all devices - desktops, laptops, tablets and cellphones, because it adapts to the size of the device on which it is being viewed.
There was a time in the earlier days of internet usage that most people used a desktop computer for surfing the internet. But times change, and laptops became widely used. To web designers this was not an issue because laptops generally had screens almost as large as those of desktops.
Later, many people discovered the convenience of using their smart-phones to access the internet. Although the screens were small, the convenience of being always available has made the cellphone very popular for surfing.
Still later along came the tablet, which was to many the ideal compromise between the portability of the cellphone and the large screen of a desktop.
The world of web design struggled to keep up. So we found many web sites being designed for only desktop users, or having two separate designs - one for desktops and one for cellphones. This of course increased both the design and the maintenance effort, and was certainly not the ideal solution.
In 2014 statistics showed that 29% of web searches were from mobile devices.
A year later Google confirmed that mobile search queries have overtaken desktop queries
Changing times indeed!
Why does this present a challenge to web designers? Well, there are a number of reasons:
Technology is catching up, and today there are technical solutions to these issues.
It is no longer a question IF you should go responsive, it is only a question of WHEN.