A responsive website is a website aimed at creating websites to provide a consitent viewing experience - easy reading and navigation with a minimum of resizing, panning, and scrolling—across a wide range of devices (from desktop computer monitors to mobile phones)
The website adapts it's layout so that content is displayed well on the device the website is opened on.
So why should your website be responsive?
20% of google searches are done on mobile and tablets. In 2012, google did a study that shows that 65% of all searches for information start on a smart phone and continue on a screen (desktop).