In order to fix this issue, you can try several things. You can try to clear the cache of the browser that you currently use to play videos. To do that, you can go to Settings -> Applications -> All tab, select the browser application and tap Clear Cache. If you use the YouTube application, also consider to clear its cache. Clear the cache and cookies of your phone. To do that, go to Settings -> Privacy and Security -> tap on Clear Cache and then on Clear all cookie data. If none of this works, try to update your Android to the later version. You can find this option in Settings -> About -> Check for updates. Rebooting the device or using a different network to connect to the Internet might help also.
