Responsive websites are websites that "respond" to their environment. Responsive websites are situated on the same domains, with the same links and same content as your desktop website but arranged to fit inside smaller (phones, tablets) or larger (projectors, large monitors) screens. This means that you don't need to post links on social media to different websites, you don't have to build a website for mobile (ever seen a website that has m. in front of it?) and you don't have to have any of the pains that are caused by managing two separate content libraries.
In short, all of our websites are responsive designs but a lot of people look for responsive design so we had to make a page to show those people that yes, we do build this way. If you need a website that's responsive you've come to the right place.