Responsive websites adjust to the size of the device you are using. If you’re on a desktop computer, you can change the size of your browser window and watch the site scale and morph dynamically, seamlessly in real time. On a smartphone or tablet device, the site has already been programmed to display at the right size. This ensures everything looks right on any screen, and eliminates the necessity to have a standalone mobile app (although you may still want a mobile app for other reasons).
The Financial Brand ran across the following responsive websites from retail banks and credit unions around the world. For each website, the browser version is shown on the left, and the condensed smartphone version is shown on the right. Bookmark this gallery and use it for inspiration.