Both responsive and adaptive design attempt to optimize the user experience across different devices, adjusting for different viewport sizes, resolutions, usage contexts, control mechanisms, and so on.
Responsive design works on the principle of flexibility - a single fluid website that can look good on any device. Responsive websites use media queries, flexible grids, and responsive images to create a user experience that flexes and changes based on a multitude of factors.
Adaptive design is more like the modern definition of progressive enhancement. Instead of one flexible design, adaptive design detects the device and other features, and then provides the appropriate feature and layout based on a predefined set of viewport sizes and other characteristics.