Responsive web design is a web design technique where the website automatically adapts based on the device. I always show people the BostonGlobe.com website when explaining responsive design. Visit the website on a desktop browser and resize the width of the browser. As you will see, the content re-prioritizes itself based on the width. The idea behind responsive web design is that a single website can provide an optimal experience on all device types. This would include smartphone, tablets, laptops and desktops.
Now the question is when should you choose a responsive design website and when should you go for a straight up mobile version of your website. Google has a comprehensive post on its blog on the subject. But here is a quick chart to help you make that decision:
I think it is better to go with a responsive design website. The extra development time and budget required will be absolutely worth it. Who wants to maintain two websites anyway?