Responsive Website Design gives your viewer the choice of browser they want to view your website on. Responsive Website Design gives that flexibility and ensures that your viewer has a pleasant interaction with your website.
Ease of navigation and less time wasting is vitally important when it comes to the SEO of your website. The major search engines have stated that they will give preference to websites that are responsive.
Wiki explains it like this:
"Responsive web design (RWD) is a web design approach 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 mobile phones to desktop computer monitors).
A site designed with RWD adapts the layout to the viewing environment by using fluid, proportion-based grids, flexible images, and CSS3 media queries, an extension of the @media rule.
Server-side components (RESS) in conjunction with client-side ones such as media queries can produce faster-loading sites for access over cellular networks and also deliver richer functionality/usability avoiding some of the pitfalls of device-side-only solutions."