Adaptive web design, mobile web design
Simply put, Responsive Web Design is like a little black dress - no matter what the occasion is, it will always look great.
But seriously, when a website is responsive it means that the layout will respond to the screen size it is being displayed upon. So on a large desktop screen the layout of a website will usually have several columns and the content will be laid out in various blocks. If you look at the same website on a mobile device the layout will "respond" to show all the content in a single column, without the loss of content or the need for a separate website.
Adaptive web design and responsive web design are mostly the same. The difference between them lies in the grid (structure) they use. A responsive layout uses a fluid grid that flows with the width of the window. An adaptive layout will have fixed points in its structure where the layout adapts to the window width.
Mobile web design refers to design for mobile devices. This might include responsive and adaptive web design, but not always.