Desktop
A desktop website is a website that is viewed on a computer. The desktop website will often contain a wide range of functions and features (graphics, video, flash, etc).
Mobile
A mobile website is a website that is viewed on a mobile device (smartphone).
Mobile sites usually include only the most crucial functions and features.
Responsive
A responsive website is a website that adapts to the device that is being used to see the website (from smartphones to desktop computer monitors).