As the number of mobile devices increase, so does the need to make a website display properly on them. Sometimes this is solved by having a main website and a mobile website which then of course requires both to be maintained.
A responsive website tackles this issue head on by having a single website for all devices, laid out so as to be easy to read and navigate with a minimum of resizing, panning and scrolling. A responsive website essentially indicates that a website has been created to use CSS3 media queries, an extension of the @media rule, with fluid proportion-based grids, to adapt the layout to the screen size.
Depending on the complexity of your existing website you could just plug-in the responsive elements directly without a re-build.