There are a couple of things not everybody notices, but once you've seen it, it just keeps annoying you. Tearing is one of those things, and appears if there is rapid movement in a movie or video game and you suddenly see the picture chopped in a couple of bands that are not aligned. The picture on the right gives an impression what screen tearing looks like. To learn a lot more about it just visit the wikipedia page (also the source of the picture).
Because of the graphics card driver or window manager not correctly aligning with the repeat rate of your monitor, you might experience tearing especially when using watching videos with youtube or your favorite media play in fullscreen. If you are using another window manager like Cinnamon or Gnome 3 you might not experience this problem. This article explains how to get your Mate Desktop by using compiz to make use of the same functionality to avoid tearing. As the other window managers the magic words are compositing and vsync. Enabling compositing alone might not do the trick, but using compiz and tweaking a bit definitely does. This HowTo fixes all tearing in connection with an intelHD or nVidia graphics chip.
Update: Scroll to the very end of this howto for fixing tearing with Ubuntu 16.04 + Mate
1. Install Compiz and Compiz Config Settings Manager
2. Set Compiz as default compositing manager for Mate Desktop
3. Configure Compiz to fix tearing
4. Some extra steps with nvidia graphics (using the standard nouveau driver)
Update for Ubuntu 16.04 + Mate
While the above steps fixed the tearing issues for a couple of versions since this howto was written, the issue disappeared for a while but seems to have come back now with the 16.04 LTS + Mate. But luckily, this time the fix is much easier, so here we go