A responsive website is a single website that can respond to the environment it’s in. For example; Desktop computer, laptop, iPad, or smart phones.
In the early days of smart phones we would build stand alone websites specific to the mobile platform. As the mobile browser was not really capable of extensive content and images this allowed us to strip it down to the bare basics. Then came tablets and we created another version of the website for tablet or optimised the existing website to work a bit better on tablet.
Fast forward to today, mobile browsers are becoming more powerful as are tablets meaning we can display more content without freezing the phone. This means we can now build a single site that automatically adjusts its layout and pieces of functionality to suit every display size. From mobile to tablet all the way to the desktop computer, your website can look the best it can on every platform.
All this means cheaper development costs for more platforms, and easier updating across all platforms. Why would you build it any other way?