Responsive websites are websites that are meant to adapt to the devices they are being displayed on. If your using a desktop, a laptop, a tablet or an smartphone your website will be able to adjust and change the content and the images in a way that is layout in a way that it makes sense. It’s going to re-position things in a way that make sens to the device that you are using.
Responsive webdesign has become a real popular trend lately because of the alternative was crappy. (getting a specific web site for a mobile platforms or tablet).