Trouble with playing videos on Facebook can usually be easily solved. Videos on Facebook insist on using Adobe Flash Player to work at all. So if your browser has problems with Flash, then that might be the source of the failure.
Try to update Flash Player to its latest version on their website. Reset your browser afterwards and it should be good to go. Speaking on resetting, Quitting and reopening your browser will usually fix most of the problem concerning the browser not working as you usually want it to be. This will also make sure that any prior updates you have done is properly installed as well. Switching the browser to something more mainstream (i.e. Internet explorer, Google Chrome, Opera, Safari, Mozilla Firefox) could help to identify if your current browser is the problem itself. If you can’t play videos at all, then it’s probably because you can’t have Flash on your computer operating system, the videos will just show up as an unclickable thumbnail in that case.
Any problem related to the apps is more troublesome, however. There is no clear way of fixing it. If you can’t view the videos or it prompts you with a notice that says “playback error”, a simple reboot should do the trick. The rebooting solution should apply too if you are viewing videos from your phone browser. Note that Apple iOS does not support flash, so their standard Safari browser can’t play Facebook videos. Uninstalling and reinstalling the Facebook apps could solve most of the issue at hand, basically allowing the apps to start fresh again. If problem persists, then stay calm and wait for the bug fix update to hit the store.