This all about a website displaying correctly on whatever device is being used to view it. Desktop, touch pad or smart phone, it could even be an e-reader.
The useability of the site is really even more important than how it looks. if the 'fat finger' that touch screen devices see can't reasonably select a link from a menu there is a problem.
Google statistics show a massive growth in the proportion of smartphone users, showing that the use of mobile users has now overtaken the use of desktop machines, Is your website up for this, most are not.
All our new websites are responsive, look good and work well on virtually any device. We are also now reviewing past projects to look to bring them up to date. No small task, there's a lot of them.