Simply put, a responsive website is a website which adjusts how it displays it's content and the layout it uses in 'response' to the capabilities of the device the website is being displayed upon..
So why would you want a responsive website? Well it used to be the case that the nearly all websites were viewed on desktop machines (PC's), but nowadays it is a complete mix of the following devices:
Traditional desktop machines, Portable or laptop machines, Smart Phones and embedded browsers (like in your TV, game console or set top box).
Each of these have a different combination of screen sizes, resolutions and capabilities. So one layout won't work well for all these different devices - hence where responsive design comes in.