Responsive designed websites are ones that automatically adapt to the screen and device they are being viewed on. This means that the text / images and layout will shrink or enlarge in proportion to each other when viewed on any device.
Most new websites designed should be mobile compatible, but if you have an older site you may find that it does not render correctly on tablets and mobiles for example. This is because of a number of factors, but do not despair, it does not automatically mean that you need a completely new site.
Depending on what content management system you use, you may be able to find a plug in or add on that will automatically create a mobile version of your site. You may have to configure this to ork properly but can save a lot of time and money. Google also takes this in to account when it comes to ranking, favouring site that are responsive.