Responsive design means designing your page to look good across a variety of devices, from large desktop monitors down to small smartphones.
Responsive design involves scaling your typefaces and images and modifying your layouts with CSS in response to the type of screen being used. This is generally preferable to other options like having a dedicated mobile site.
Responsive design is typically achieved with CSS tools like flex box and media queries.
As of 2015, 56% of web traffic is done through a mobile devices, so responsive design is generally not optional.
Code Academy has a free responsive web design course that can walk you through the basics.