What is Responsive Design?
Responsive web design (RWD) is an approach to web design aimed at crafting sites to provide an optimal 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).
Why is it important?
First and foremost, you want vistors to your site to get the most out of it, and have the best experience possible. Your site being able to automatcially, in essence, re-format itself to give the visitor the best experience, will help to keep visitors on your site. The easier your site is to view on whatever device the visitor is using, the more likely they are to spend more time on your site and, if you selling things, make a purchase.
It's also easier to maintain your site.
With the proliferation of mobile devices becoming the defacto 'device of choice' people are using to browse the Internet, up until now, a vast number of sites actually maintain two sites; one for desktop visitors, the other for mobile device visitors. Each one is formatted for a different screen size. A 'script' is used when the visitor lands on one of your pages that is able to determine the size of the screen they're using and send them to the proper portion of your site that is optimized for their screen size.
What this means is that you are actually maintaining two of just about everything; pages, pictures, etc.; each one being a different layout and/or size to accommodate the varying screen sizes.
Responsive Design solves these problems
To make a web page work in a browser, there are 'codes'. You probably heard of HTML (hypertext markup language). These are codes a web broswer understands and acts on to do things like make text bold, or italic and thing of that nature.
There are also codes built into the latest version of HTML (HTML5). These codes are understood by most up-to-date browsers and they allow browsers to automatically 're-format' the site's content to fit the screen size of the device they're being viewed on.
Search engine rankings
If they haven't already heard about it, search sites (such as Google and Bing) will be taking into account whether or not your site has the capability of either re-formatting itself to accommodate various screen sizes or offer a different view for different screen sized devices (like desktop and cell phone).
If you don't offer either, it will lower your rankings in these search engines, perhaps dramatically, and your website (and your online business) could suffer greatly.
Responsive Design solves these problems.