There are comprehensive posts and wiki articles explaining the topic of “responsive design”, so I’ll simply sum it up by saying:
Responsive design makes sure your website is viewable / readable on whatever device or screen size the site visitor is using.
I have found that most WordPress themes these days come responsive “out of the box”, but some remain (for various reasons) as a one-size-fits-all-screens layout.
The most pressing screen size requiring responsive design elements that I’ve worked into quite a few client sites is the mobile browser for standard-sized phones. With the exception of some phablets, most phones and similarly sized, web-enabled devices require a lot of zoom to read only small portions of a webpage. So, my most concerted efforts have been to make screen views with a width of less than ~580px responsive.
A cool web-tool I sometimes utilize is Responsinator. There are a number of similar sites that will do this for you. As more and more people use their phones (and emerging technologies along that same line), mobile sites will become more and more mainstream.
Something to consider now and in the near future!