You cannot turn a web design magazine a page these days without seeing an article on responsive design. There is a huge amount of thought and discussion going about this topic and it is an exciting development in the market.
Responsive design means that your website is able to be presented in different media whether it be a standard browser, a mobile device or a tablet. It is hugely desirable if your client base is mobile and demands that your site be seen easily in these mediums.
However it does take a lot of extra design hours and testing and there will always be another device or browser that may not present your content exactly right.
Is it important for your site?
The quick answer is probably yes but you need to look at your current and prospective client base before investing in the time and energy. Your Google Analytics data can show you where your current traffic is coming from and this can fuel the time and investment that you put into a design. For new sites you need to spend some additional time on your design package to establish the right criteria for what is presented on the different media.
We love responsive and themes that reflect the different media and as mobile grows it will become a standard part of packages but for scope and budget planning you should think of client base ad future expansion when doing the design planning phase.