Responsive Web Design: What It Is and How To Use It is a nice explanation of the responsive design concept as well as a pretty thorough examination of how to implement responsive design. (Smashing Magazine)
Media queries can be used to assign different CSS values to elements depending on the screen size. (Mozilla Developer Network)
Another resource on using media queries from the folks at Google. (Google Developers)
The Responsive Web Design Testing Tool allows you to see how your site will appear in different screen sizes simultaneously. (Matt Kersley)
Responsive Design Fundamentals is an online course developed by Google to teach the basics of responsive design. It is planned as a two-week course comprising roughly 12 hours. (Udacity)
The responsive design built into the CSS of Bootstrap allows for the flow of elements using columns. This site is built using Bootstrap. (Bootstrap)
The blog post that started it all- why we should be designing for mobile. (Luke Wroblewski
Mobile First Design: Why It’s Great and Why It Sucks offers a brief examination of the philosophy and covers some of the problems designers face in trying to implement it. (Joshua Johnson, Design Shack)