Responsive web design (often abbreviated to RWD) is a web design approach aimed at building sites to provide an optimal viewing experience. It offers easy reading and navigation with a minimum of resizing, panning, and scrolling—across a wide range of devices (from desktop computer monitors to mobile phones).
It is popular because of the proliferation of hardware devices accessing websites. There are hundreds of different types of smartphones, tablets, and computers with different screen sizes and characteristics.
In simple terms, a responsive web design uses "media queries" to figure out what resolution of device it's being served on. Flexible images and fluid grids then size correctly to fit the screen. You can also do device detection to, for example, enable swiping.
The ultimate aim is to build a website once, and, if done correctly, it works seamlessly across thousands of different screens.