So, what is a 'Responsive' website?
To me Responsive design means simply making your website mobile compatible. There is some debate between the methods used, Responsive vs Adaptive vs discreet mobile versions. Of course, none of this matters to you or your customers at all. All you really need to know, is that your Responsive website should look great on your mobile too!
A responsive design starts at desktop size, then collapses down as the screen size reduces and/or changes shape (eg. rotating your phone or tablet). This means layouts are percentage based, rather than the old-school fixed width of 960 pixels which you were used to seeing through the 2000's. The other major area which needs attention is the differences in usability between a desktop and a touch device. Gestures, for example.
People browsing the web from their smartphone/tablet is increasing. If you look at the Google/TNS 2013 statistics for New Zealand smartphone usage you will see that 97% of smartphone owners use their device to access the web! And mobile devices have actually overtaken desktop/laptop in frequency of use over once per day. The other interesting information was where people are surfing from on their smartphones. 'On the go/While walking' stuck out as previously un-tapped in the days of primarily desktop browsing.
These days, I recommend to all of my new customers to create their website Responsive. If it's something you can do, it can only make your information more accessible to more people more of the time.