When a website is responsive, the layout and/or content responds (or, adapts) based on the size of the screen it’s presented on.
Typically, there have been four general screen sizes that responsive design has been aimed at: the widescreen desktop monitor, the smaller desktop (or laptop), the tablet, and the mobile phone.
In short, you (the digital publisher, developer, and designer) should care because you want the visitors to your website to have the best experience possible, without forcing them to adapt themselves.
via A Beginner’s Guide to Mobile Responsive Design | Copyblogger.