With the web changing at a rapid pace a website redesign can be an intimidating undertaking. How can you build a website that will look great and work well on current and future devices? Mobile, tablets, desktop, laptops, TVs, etc. There is only one relatively safe bet on future proofing your website and it's responsive web design or RWD.
A responsive website automatically resizes and reformats to browser dimensions and device functionality. So on a desktop it will look like a "normal" website but on a phone it will look and function like a mobile optimized site. Often a responsive website transitions from a mouse-friendly to a touch-friendly interface. A phone number becomes a "Call" button, etc.
We recently worked with Angie's List to redesign many of their landing pages to be responsive. For a live example go to this page and if you are on a desktop/laptop computer you can drag the bottom right corner of the browser to see the page respond to the new dimensions. When these pages relaunched with the responsive design Angie's List saw a 40% increase in conversions! We also recently launched a responsive website for Tyson Onsite that demonstrates a full responsive website. And there's the Powderkeg website we did for the recent Verge conference here in Indy.
In the coming weeks we will be relaunching our own site with a responsive front end. Our current website (which you are on right now) is mobile optimized which is a decent halfway solution and the only one available to us when we built it 3 years ago (Responsive became a "thing" in 2011). I'd be happy to discuss the difference between Responsive and mobile optimized in the comments if anyone wants clarification on the differences.
Responsive design future proofs your site against devices that haven't even been dreamed up. With mobile and tablet usage skyrocketing (many sites we monitor as quickly heading to 50% mobile/tablet traffic) it's clear that a website built to last has to be built to be responsive. Also, Google recently came out supporting Responsive as the gold standard which means it most likely has SEO implications as well. So if you've been waiting for mobile to settle down, it has, get responsive or get left behind.
Read more about responsive web design on Wikipedia