Responsive websites are extremely important as they need to be built to adapt to the device in which it is being view on. With many people now using phones and tablets, it is vital that your website displays and functions correctly on a number of different screen sizes.
If you already have a website and it is not responsive then you may be able to get a programmer to alter it for you, but this may be costly as its not always an easy change and may require many elements changing to get it all to work. If your site uses a theme from somewhere such as WordPress or Drupal then you can often find responsive themes which can be uploaded to the server. You can then swap your website over to use the new theme, but you still may need to do a few changes to get it to look as it should.
Google has actually started to use this as part of their algorithm (along with a number of other items) and is favouring responsive sites over non responsive ones to allow them to give the visitor the best possibly experience.