Responsive web design as defined on Wikipedia is as follows
Responsive web design (RWD) is an approach to web design aimed at crafting sites to provide an optimal viewing and interaction experience—easy reading and navigation with a minimum of re sizing, panning, and scrolling—across a wide range of devices (from desktop computer monitors to mobile phones)
After the surge in the usage of mobile phones, tablets and other hand held devices, websites that had earlier been designed with desktops or large screens in mind are now being re-designed to display properly on the small screens. As you are all aware that people now browse more on their mobile phones using WiFi or the internet packages offered by their respective carriers the need for being responsive has increased ever more.
More so after Google updated their algorithm last year to give a boost to mobile friendly websites and web pages. Google will now give more importance to those results which are mobile friendly so if your website is not mobile friendly then you may not come up in Google mobile search results and lose considerable traffic.
Google has even launched a tool known as the Mobile Friendly Test Tool which will tell you if your current website is mobile friendly or not.