It really started to take hold in 2012, and by now is an absolute must for your website. Basically it means regardless of what sort of device your site is being seen on; a tablet, phone, laptop or desktop, it looks great. The text is readable, even the layout adjusts (or responds, see how that works!?) to suit the display.
I just created this site today (so the client still needs to populate the content a fair bit), but you can see from the picture above it looks just as nice on mobile as a full screen.
I love web design.