The idea behind responsive web design is to give developers the ability to make a single website look good on every device. This is becoming increasingly important due to the amount of searches being made on mobile devices. Google says there are more searches taking place on smartphones and tablets than on desktops in 10 countries. This number has since increased making it impossible to ignore the need for a "responsive" website.
When users pick up their smartphones and make a search for products or services that your business offers you want your website to be visible. Without a responsive website, customers can become easily frustrated while moving on to the next opportunity causing you to lose leads and sales. There are also search engine optimization penalties which could cause your website not to rank on mobile devices.
There are a few factors that affect the responsiveness of a website, however, the one most ubiquitous and the one I will elaborate on is a media query. A media query is code that tells the browser how to render content within a document based on certain variables that are defined (e.g. media, width, height, color, etc.).