Google has just moved responsive web design from "Important" to "Necessary" with their announcement that they are putting more weight on sites being mobile friendly. Google says the effect of this change will be "profound." What does this mean? If your site displays properly on all device types and sizes, you get a higher rank in Google search results.
Google warned webmasters 2 months ago that this was coming (not a normal practice for Google) which tells us that it carries a lot of weight. Google says recommends that "webmasters follow the industry best practice of using web design, namely serving the same HTML for all devices, using media queries to decide rendering on each device."
So how can you tell if your website is responsive? The easiest way is to look at it on a mobile device. Do you have to "zoom in" to read the information? If you do, then your site is not responsive.
If you don't have a mobile device, you can also use your browser to check. In Firefox you can use the shortcut [ctrl][shift]M to switch to "Responsive Design Mode" which will enable you to select from a variety of screen sizes. Just browse to your site, hit "[ctrl][shift]M" and choose different screen sizes from the drop down menu at the top. Your site should be usable with all of the sizes.
If your site is not responsive, the best way to fix that is with a new site design. It is very difficult to turn a non-responsive website into a responsive one.