Responsive design is the latest incarnation of what used to be different designs for different interfaces. Previously there was the desktop and mobile interfaces. Webmasters designed and maintained two different websites for each. It was double the work and usually ended up with very ugly mobile websites. With the advent smartphones (both small screen and large), different sized tables and large screen desktops, it is no longer feasible to have different designs for each form factor. This is where responsive design came in.
The first and most important aspect of responsive design is that there should be a very limited set of resolutions that the website adjusts to. Those are the larger changes, from phone to tablet to screen. Any size in between must be adjusted to automatically with fluid widths and elements. Done properly, on a desktop you could just resize the window horizontally and it should adjust as you shrink the window without any elements going out of place.
The second is navigation. A good responsive website should hide the navigation bar and create a touch based drop down icon for the menu. This should happen at the size just below tablet portrait width. Real estate on a small screen is limited and it is not acceptable to have anything other than a single button with the logo.
The third characteristic is simplicity. All of these changes need to happen quickly. Sites that take longer to load and render will have issues with search engines and users alike. The more complicated the design, the slower it will be to adjust and even to scroll on a mobile browser.