The term responsive web design was coined by Ethan Marcotte. In his seminal List Apart article (http://www.alistapart.com/articles/responsive-web-design/) he consolidated three existing techniques (flexible grid layout, flexible images, and media and media queries) into one unified approach and named it responsive web design. The term is often used to infer the same meaning as a number of other descriptions such as fluid design, elastic layout, rubber layout, liquid design, adaptive layout, cross-device design, and flexible design.
To name just a few! However, as Mr. Marcotte and others have eloquently argued, a truly responsive methodology is actually more than merely altering the layout of a site based upon viewport sizes. Instead, it is to invert our entire current approach to web design. Instead of beginning with a fixed width desktop site design and scaling it down and re-flowing the content for smaller viewports, we should design for the smallest viewport first and then progressively enhance the design and content for larger viewports.
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Responsive web design in a nutshell
To attempt to put the philosophy of responsive web design in a nutshell, I would say it's the presentation of content in the most accessible manner for any viewport that accesses it. Conversely, a truly "mobile website" is needed when an experience requires specific content and functionality based upon the device accessing it. In these cases, a mobile website presents an entirely different user experience to its desktop equivalent.