Freesync and GSync both provide an alternative to V Sync. To explain why it's important, we have to explain why Vsync is deficient.
So with Vsync, essentially, your card slows itself down in order to avoid going faster than your monitor, which is always 60hz/fps. In doing so, it adds latency and can only run at either 60fps or 30fps, because your monitor is dictating what your video card will do.
Freesync and G Sync flip that around, and have your video card tell your monitor what to do, and eliminates tearing by making the monitor display whatever the video card is ready for. This reduces latency, and also means that you won't stutter in between 60 and 30 fps... So as your FPS fluctuates from say 45fps to 60, it'll stay smooth the whole range through, as the monitor will adaptively, natively, support all of those framerates, instead of just trying to make everything work at 60.
Now the main difference between Gsync and Freesync, is that Gsync gets a little bit wider range of fps/hz that it will work at vs Freesync (so Freesync will work from 45-144hz, and maybe GSync will work from 40-144hz), but is locked down to nVidia cards only, and works only on special monitors that have a $100 chip in it. So a Gsync monitor will be $100 more usually than it's Freesync equivalent, and will only work on nVidia cards.
Freesync in theory works on everything, as it's just a part of the standard, but no consoles yet support it, and nVidia actively blocks it, so for now it's sort of exclusive to AMD. Please note; the Xbox One X will support Freesync. No Gsync tho!
This only matters if you need the best experience, and if you're just getting into things, I wouldn't worry about it too much tbh. But Freesync will be supported by more and more devices, and Gsync only nVidia cards ever. It can effectively lock you into one card vendor atm if you buy a monitor specifically for this feature though.
