Responsive design is designing a website so that the content flows to fill the screen regardless of the screen size. When the screen size changes, the content flows dynamically to fill the screen, moving the elements as necessary to allow the webpage to be viewed on any size screen with no horizontal scrollbars.
Reactive Design is a type of website design that takes a set width point that is assigned to a page, and snaps the content to fix inside it as it changes. It is not fluid like Responsive design is, rather, as the screen size changes, it sees a certain point, and snaps the content to fit inside of it at specific size dimensions assigned for each format. For example, designer could say, for PCs, size the site to 1200 pixels wide; for tablets, size the site to 500 pixels wide; for mobile devices, size the site to 150 pixels wide. When the viewer views the site, it will snap to fit these predefined sizes, rather than dynamically resizing to fit the screen. Adaptive is another word for Reactive design. A great example of how this works can be seen here: http://blog.froont.com/9-basic-principles-of-responsive-web-design/.
Mobile first is a design concept where you begin, from the very planning stages of your website design, to design a website that works flawlessly on mobile devices. From that bas site, you then scale it up for larger screens and desktop systems. It's based on the concept that most internet browsing occurs on cell phones, and serving content that is assessible to those users is the highest priority. I don't necessaryily agree with that, but I have been on some websites that were frustratingly difficult to do anything on with a mobile device, so I definitely want to consider these viewers when designing.Links: